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Metamath Proof Explorer

Theorem bnj1398

Description: Technical lemma for bnj60 . This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

Ref Expression
Hypotheses bnj1398.1 ⊢ B = d | d ⊆ A ∧ ∀ x ∈ d pred x A R ⊆ d
bnj1398.2 ⊢ Y = x f ↾ pred x A R
bnj1398.3 ⊢ C = f | ∃ d ∈ B f Fn d ∧ ∀ x ∈ d f ⁡ x = G ⁡ Y
bnj1398.4 ⊢ τ ↔ f ∈ C ∧ dom ⁡ f = x ∪ trCl x A R
bnj1398.5 ⊢ D = x ∈ A | ¬ ∃ f τ
bnj1398.6 ⊢ ψ ↔ R FrSe A ∧ D ≠ ∅
bnj1398.7 ⊢ χ ↔ ψ ∧ x ∈ D ∧ ∀ y ∈ D ¬ y R x
bnj1398.8 No typesetting found for |- ( ta' <-> [. y / x ]. ta ) with typecode |-
bnj1398.9 No typesetting found for |- H = { f | E. y e. _pred ( x , A , R ) ta' } with typecode |-
bnj1398.10 ⊢ P = ⋃ H
bnj1398.11 ⊢ θ ↔ χ ∧ z ∈ ⋃ y ∈ pred x A R y ∪ trCl y A R
bnj1398.12 ⊢ η ↔ θ ∧ y ∈ pred x A R ∧ z ∈ y ∪ trCl y A R
Assertion bnj1398 ⊢ χ → ⋃ y ∈ pred x A R y ∪ trCl y A R = dom ⁡ P

Proof

Step Hyp Ref Expression
1 bnj1398.1 ⊢ B = d | d ⊆ A ∧ ∀ x ∈ d pred x A R ⊆ d
2 bnj1398.2 ⊢ Y = x f ↾ pred x A R
3 bnj1398.3 ⊢ C = f | ∃ d ∈ B f Fn d ∧ ∀ x ∈ d f ⁡ x = G ⁡ Y
4 bnj1398.4 ⊢ τ ↔ f ∈ C ∧ dom ⁡ f = x ∪ trCl x A R
5 bnj1398.5 ⊢ D = x ∈ A | ¬ ∃ f τ
6 bnj1398.6 ⊢ ψ ↔ R FrSe A ∧ D ≠ ∅
7 bnj1398.7 ⊢ χ ↔ ψ ∧ x ∈ D ∧ ∀ y ∈ D ¬ y R x
8 bnj1398.8 Could not format ( ta' <-> [. y / x ]. ta ) : No typesetting found for |- ( ta' <-> [. y / x ]. ta ) with typecode |-
9 bnj1398.9 Could not format H = { f | E. y e. _pred ( x , A , R ) ta' } : No typesetting found for |- H = { f | E. y e. _pred ( x , A , R ) ta' } with typecode |-
10 bnj1398.10 ⊢ P = ⋃ H
11 bnj1398.11 ⊢ θ ↔ χ ∧ z ∈ ⋃ y ∈ pred x A R y ∪ trCl y A R
12 bnj1398.12 ⊢ η ↔ θ ∧ y ∈ pred x A R ∧ z ∈ y ∪ trCl y A R
13 df-iun ⊢ ⋃ y ∈ pred x A R y ∪ trCl y A R = z | ∃ y ∈ pred x A R z ∈ y ∪ trCl y A R
14 13 bnj1436 ⊢ z ∈ ⋃ y ∈ pred x A R y ∪ trCl y A R → ∃ y ∈ pred x A R z ∈ y ∪ trCl y A R
15 11 14 simplbiim ⊢ θ → ∃ y ∈ pred x A R z ∈ y ∪ trCl y A R
16 nfv ⊢ Ⅎ y ψ
17 nfv ⊢ Ⅎ y x ∈ D
18 nfra1 ⊢ Ⅎ y ∀ y ∈ D ¬ y R x
19 16 17 18 nf3an ⊢ Ⅎ y ψ ∧ x ∈ D ∧ ∀ y ∈ D ¬ y R x
20 7 19 nfxfr ⊢ Ⅎ y χ
21 nfiu1 ⊢ Ⅎ _ y ⋃ y ∈ pred x A R y ∪ trCl y A R
22 21 nfcri ⊢ Ⅎ y z ∈ ⋃ y ∈ pred x A R y ∪ trCl y A R
23 20 22 nfan ⊢ Ⅎ y χ ∧ z ∈ ⋃ y ∈ pred x A R y ∪ trCl y A R
24 11 23 nfxfr ⊢ Ⅎ y θ
25 24 nf5ri ⊢ θ → ∀ y θ
26 15 12 25 bnj1521 ⊢ θ → ∃ y η
27 nfv ⊢ Ⅎ f R FrSe A
28 nfe1 ⊢ Ⅎ f ∃ f τ
29 28 nfn ⊢ Ⅎ f ¬ ∃ f τ
30 nfcv ⊢ Ⅎ _ f A
31 29 30 nfrabw ⊢ Ⅎ _ f x ∈ A | ¬ ∃ f τ
32 5 31 nfcxfr ⊢ Ⅎ _ f D
33 nfcv ⊢ Ⅎ _ f ∅
34 32 33 nfne ⊢ Ⅎ f D ≠ ∅
35 27 34 nfan ⊢ Ⅎ f R FrSe A ∧ D ≠ ∅
36 6 35 nfxfr ⊢ Ⅎ f ψ
37 32 nfcri ⊢ Ⅎ f x ∈ D
38 nfv ⊢ Ⅎ f ¬ y R x
39 32 38 nfralw ⊢ Ⅎ f ∀ y ∈ D ¬ y R x
40 36 37 39 nf3an ⊢ Ⅎ f ψ ∧ x ∈ D ∧ ∀ y ∈ D ¬ y R x
41 7 40 nfxfr ⊢ Ⅎ f χ
42 nfv ⊢ Ⅎ f z ∈ ⋃ y ∈ pred x A R y ∪ trCl y A R
43 41 42 nfan ⊢ Ⅎ f χ ∧ z ∈ ⋃ y ∈ pred x A R y ∪ trCl y A R
44 11 43 nfxfr ⊢ Ⅎ f θ
45 nfv ⊢ Ⅎ f y ∈ pred x A R
46 nfv ⊢ Ⅎ f z ∈ y ∪ trCl y A R
47 44 45 46 nf3an ⊢ Ⅎ f θ ∧ y ∈ pred x A R ∧ z ∈ y ∪ trCl y A R
48 12 47 nfxfr ⊢ Ⅎ f η
49 48 nf5ri ⊢ η → ∀ f η
50 11 simplbi ⊢ θ → χ
51 12 50 bnj835 ⊢ η → χ
52 12 simp2bi ⊢ η → y ∈ pred x A R
53 1 2 3 4 5 6 7 8 bnj1388 Could not format ( ch -> A. y e. _pred ( x , A , R ) E. f ta' ) : No typesetting found for |- ( ch -> A. y e. _pred ( x , A , R ) E. f ta' ) with typecode |-
54 rsp Could not format ( A. y e. _pred ( x , A , R ) E. f ta' -> ( y e. _pred ( x , A , R ) -> E. f ta' ) ) : No typesetting found for |- ( A. y e. _pred ( x , A , R ) E. f ta' -> ( y e. _pred ( x , A , R ) -> E. f ta' ) ) with typecode |-
55 53 54 syl Could not format ( ch -> ( y e. _pred ( x , A , R ) -> E. f ta' ) ) : No typesetting found for |- ( ch -> ( y e. _pred ( x , A , R ) -> E. f ta' ) ) with typecode |-
56 51 52 55 sylc Could not format ( et -> E. f ta' ) : No typesetting found for |- ( et -> E. f ta' ) with typecode |-
57 49 56 bnj596 Could not format ( et -> E. f ( et /\ ta' ) ) : No typesetting found for |- ( et -> E. f ( et /\ ta' ) ) with typecode |-
58 1 2 3 4 8 bnj1373 Could not format ( ta' <-> ( f e. C /\ dom f = ( { y } u. _trCl ( y , A , R ) ) ) ) : No typesetting found for |- ( ta' <-> ( f e. C /\ dom f = ( { y } u. _trCl ( y , A , R ) ) ) ) with typecode |-
59 58 simplbi Could not format ( ta' -> f e. C ) : No typesetting found for |- ( ta' -> f e. C ) with typecode |-
60 59 adantl Could not format ( ( et /\ ta' ) -> f e. C ) : No typesetting found for |- ( ( et /\ ta' ) -> f e. C ) with typecode |-
61 58 simprbi Could not format ( ta' -> dom f = ( { y } u. _trCl ( y , A , R ) ) ) : No typesetting found for |- ( ta' -> dom f = ( { y } u. _trCl ( y , A , R ) ) ) with typecode |-
62 rspe ⊢ y ∈ pred x A R ∧ dom ⁡ f = y ∪ trCl y A R → ∃ y ∈ pred x A R dom ⁡ f = y ∪ trCl y A R
63 52 61 62 syl2an Could not format ( ( et /\ ta' ) -> E. y e. _pred ( x , A , R ) dom f = ( { y } u. _trCl ( y , A , R ) ) ) : No typesetting found for |- ( ( et /\ ta' ) -> E. y e. _pred ( x , A , R ) dom f = ( { y } u. _trCl ( y , A , R ) ) ) with typecode |-
64 9 eqabri Could not format ( f e. H <-> E. y e. _pred ( x , A , R ) ta' ) : No typesetting found for |- ( f e. H <-> E. y e. _pred ( x , A , R ) ta' ) with typecode |-
65 58 rexbii Could not format ( E. y e. _pred ( x , A , R ) ta' <-> E. y e. _pred ( x , A , R ) ( f e. C /\ dom f = ( { y } u. _trCl ( y , A , R ) ) ) ) : No typesetting found for |- ( E. y e. _pred ( x , A , R ) ta' <-> E. y e. _pred ( x , A , R ) ( f e. C /\ dom f = ( { y } u. _trCl ( y , A , R ) ) ) ) with typecode |-
66 r19.42v ⊢ ∃ y ∈ pred x A R f ∈ C ∧ dom ⁡ f = y ∪ trCl y A R ↔ f ∈ C ∧ ∃ y ∈ pred x A R dom ⁡ f = y ∪ trCl y A R
67 64 65 66 3bitri ⊢ f ∈ H ↔ f ∈ C ∧ ∃ y ∈ pred x A R dom ⁡ f = y ∪ trCl y A R
68 60 63 67 sylanbrc Could not format ( ( et /\ ta' ) -> f e. H ) : No typesetting found for |- ( ( et /\ ta' ) -> f e. H ) with typecode |-
69 12 simp3bi ⊢ η → z ∈ y ∪ trCl y A R
70 69 adantr Could not format ( ( et /\ ta' ) -> z e. ( { y } u. _trCl ( y , A , R ) ) ) : No typesetting found for |- ( ( et /\ ta' ) -> z e. ( { y } u. _trCl ( y , A , R ) ) ) with typecode |-
71 61 adantl Could not format ( ( et /\ ta' ) -> dom f = ( { y } u. _trCl ( y , A , R ) ) ) : No typesetting found for |- ( ( et /\ ta' ) -> dom f = ( { y } u. _trCl ( y , A , R ) ) ) with typecode |-
72 70 71 eleqtrrd Could not format ( ( et /\ ta' ) -> z e. dom f ) : No typesetting found for |- ( ( et /\ ta' ) -> z e. dom f ) with typecode |-
73 68 72 jca Could not format ( ( et /\ ta' ) -> ( f e. H /\ z e. dom f ) ) : No typesetting found for |- ( ( et /\ ta' ) -> ( f e. H /\ z e. dom f ) ) with typecode |-
74 57 73 bnj593 ⊢ η → ∃ f f ∈ H ∧ z ∈ dom ⁡ f
75 df-rex ⊢ ∃ f ∈ H z ∈ dom ⁡ f ↔ ∃ f f ∈ H ∧ z ∈ dom ⁡ f
76 74 75 sylibr ⊢ η → ∃ f ∈ H z ∈ dom ⁡ f
77 10 dmeqi ⊢ dom ⁡ P = dom ⁡ ⋃ H
78 9 bnj1317 ⊢ w ∈ H → ∀ f w ∈ H
79 78 bnj1400 ⊢ dom ⁡ ⋃ H = ⋃ f ∈ H dom ⁡ f
80 77 79 eqtri ⊢ dom ⁡ P = ⋃ f ∈ H dom ⁡ f
81 80 eleq2i ⊢ z ∈ dom ⁡ P ↔ z ∈ ⋃ f ∈ H dom ⁡ f
82 eliun ⊢ z ∈ ⋃ f ∈ H dom ⁡ f ↔ ∃ f ∈ H z ∈ dom ⁡ f
83 81 82 bitri ⊢ z ∈ dom ⁡ P ↔ ∃ f ∈ H z ∈ dom ⁡ f
84 76 83 sylibr ⊢ η → z ∈ dom ⁡ P
85 26 84 bnj593 ⊢ θ → ∃ y z ∈ dom ⁡ P
86 nfre1 Could not format F/ y E. y e. _pred ( x , A , R ) ta' : No typesetting found for |- F/ y E. y e. _pred ( x , A , R ) ta' with typecode |-
87 86 nfab Could not format F/_ y { f | E. y e. _pred ( x , A , R ) ta' } : No typesetting found for |- F/_ y { f | E. y e. _pred ( x , A , R ) ta' } with typecode |-
88 9 87 nfcxfr ⊢ Ⅎ _ y H
89 88 nfuni ⊢ Ⅎ _ y ⋃ H
90 10 89 nfcxfr ⊢ Ⅎ _ y P
91 90 nfdm ⊢ Ⅎ _ y dom ⁡ P
92 91 nfcrii ⊢ z ∈ dom ⁡ P → ∀ y z ∈ dom ⁡ P
93 85 92 bnj1397 ⊢ θ → z ∈ dom ⁡ P
94 11 93 sylbir ⊢ χ ∧ z ∈ ⋃ y ∈ pred x A R y ∪ trCl y A R → z ∈ dom ⁡ P
95 94 ex ⊢ χ → z ∈ ⋃ y ∈ pred x A R y ∪ trCl y A R → z ∈ dom ⁡ P
96 95 ssrdv ⊢ χ → ⋃ y ∈ pred x A R y ∪ trCl y A R ⊆ dom ⁡ P
97 simpr ⊢ f ∈ H ∧ z ∈ dom ⁡ f → z ∈ dom ⁡ f
98 67 simprbi ⊢ f ∈ H → ∃ y ∈ pred x A R dom ⁡ f = y ∪ trCl y A R
99 98 adantr ⊢ f ∈ H ∧ z ∈ dom ⁡ f → ∃ y ∈ pred x A R dom ⁡ f = y ∪ trCl y A R
100 r19.42v ⊢ ∃ y ∈ pred x A R z ∈ dom ⁡ f ∧ dom ⁡ f = y ∪ trCl y A R ↔ z ∈ dom ⁡ f ∧ ∃ y ∈ pred x A R dom ⁡ f = y ∪ trCl y A R
101 eleq2 ⊢ dom ⁡ f = y ∪ trCl y A R → z ∈ dom ⁡ f ↔ z ∈ y ∪ trCl y A R
102 101 biimpac ⊢ z ∈ dom ⁡ f ∧ dom ⁡ f = y ∪ trCl y A R → z ∈ y ∪ trCl y A R
103 102 reximi ⊢ ∃ y ∈ pred x A R z ∈ dom ⁡ f ∧ dom ⁡ f = y ∪ trCl y A R → ∃ y ∈ pred x A R z ∈ y ∪ trCl y A R
104 100 103 sylbir ⊢ z ∈ dom ⁡ f ∧ ∃ y ∈ pred x A R dom ⁡ f = y ∪ trCl y A R → ∃ y ∈ pred x A R z ∈ y ∪ trCl y A R
105 97 99 104 syl2anc ⊢ f ∈ H ∧ z ∈ dom ⁡ f → ∃ y ∈ pred x A R z ∈ y ∪ trCl y A R
106 105 rexlimiva ⊢ ∃ f ∈ H z ∈ dom ⁡ f → ∃ y ∈ pred x A R z ∈ y ∪ trCl y A R
107 eliun ⊢ z ∈ ⋃ y ∈ pred x A R y ∪ trCl y A R ↔ ∃ y ∈ pred x A R z ∈ y ∪ trCl y A R
108 106 83 107 3imtr4i ⊢ z ∈ dom ⁡ P → z ∈ ⋃ y ∈ pred x A R y ∪ trCl y A R
109 108 ssriv ⊢ dom ⁡ P ⊆ ⋃ y ∈ pred x A R y ∪ trCl y A R
110 109 a1i ⊢ χ → dom ⁡ P ⊆ ⋃ y ∈ pred x A R y ∪ trCl y A R
111 96 110 eqssd ⊢ χ → ⋃ y ∈ pred x A R y ∪ trCl y A R = dom ⁡ P