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Metamath Proof Explorer

Theorem bnj1467

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 bnj1467.1 ⊢ B = d | d ⊆ A ∧ ∀ x ∈ d pred x A R ⊆ d
bnj1467.2 ⊢ Y = x f ↾ pred x A R
bnj1467.3 ⊢ C = f | ∃ d ∈ B f Fn d ∧ ∀ x ∈ d f ⁡ x = G ⁡ Y
bnj1467.4 ⊢ τ ↔ f ∈ C ∧ dom ⁡ f = x ∪ trCl x A R
bnj1467.5 ⊢ D = x ∈ A | ¬ ∃ f τ
bnj1467.6 ⊢ ψ ↔ R FrSe A ∧ D ≠ ∅
bnj1467.7 ⊢ χ ↔ ψ ∧ x ∈ D ∧ ∀ y ∈ D ¬ y R x
bnj1467.8 No typesetting found for |- ( ta' <-> [. y / x ]. ta ) with typecode |-
bnj1467.9 No typesetting found for |- H = { f | E. y e. _pred ( x , A , R ) ta' } with typecode |-
bnj1467.10 ⊢ P = ⋃ H
bnj1467.11 ⊢ Z = x P ↾ pred x A R
bnj1467.12 ⊢ Q = P ∪ x G ⁡ Z
Assertion bnj1467 ⊢ w ∈ Q → ∀ d w ∈ Q

Proof

Step Hyp Ref Expression
1 bnj1467.1 ⊢ B = d | d ⊆ A ∧ ∀ x ∈ d pred x A R ⊆ d
2 bnj1467.2 ⊢ Y = x f ↾ pred x A R
3 bnj1467.3 ⊢ C = f | ∃ d ∈ B f Fn d ∧ ∀ x ∈ d f ⁡ x = G ⁡ Y
4 bnj1467.4 ⊢ τ ↔ f ∈ C ∧ dom ⁡ f = x ∪ trCl x A R
5 bnj1467.5 ⊢ D = x ∈ A | ¬ ∃ f τ
6 bnj1467.6 ⊢ ψ ↔ R FrSe A ∧ D ≠ ∅
7 bnj1467.7 ⊢ χ ↔ ψ ∧ x ∈ D ∧ ∀ y ∈ D ¬ y R x
8 bnj1467.8 Could not format ( ta' <-> [. y / x ]. ta ) : No typesetting found for |- ( ta' <-> [. y / x ]. ta ) with typecode |-
9 bnj1467.9 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 bnj1467.10 ⊢ P = ⋃ H
11 bnj1467.11 ⊢ Z = x P ↾ pred x A R
12 bnj1467.12 ⊢ Q = P ∪ x G ⁡ Z
13 nfcv ⊢ Ⅎ _ d pred x A R
14 nfcv ⊢ Ⅎ _ d y
15 nfre1 ⊢ Ⅎ d ∃ d ∈ B f Fn d ∧ ∀ x ∈ d f ⁡ x = G ⁡ Y
16 15 nfab ⊢ Ⅎ _ d f | ∃ d ∈ B f Fn d ∧ ∀ x ∈ d f ⁡ x = G ⁡ Y
17 3 16 nfcxfr ⊢ Ⅎ _ d C
18 17 nfcri ⊢ Ⅎ d f ∈ C
19 nfv ⊢ Ⅎ d dom ⁡ f = x ∪ trCl x A R
20 18 19 nfan ⊢ Ⅎ d f ∈ C ∧ dom ⁡ f = x ∪ trCl x A R
21 4 20 nfxfr ⊢ Ⅎ d τ
22 14 21 nfsbcw ⊢ Ⅎ d [˙y / x]˙ τ
23 8 22 nfxfr Could not format F/ d ta' : No typesetting found for |- F/ d ta' with typecode |-
24 13 23 nfrexw Could not format F/ d E. y e. _pred ( x , A , R ) ta' : No typesetting found for |- F/ d E. y e. _pred ( x , A , R ) ta' with typecode |-
25 24 nfab Could not format F/_ d { f | E. y e. _pred ( x , A , R ) ta' } : No typesetting found for |- F/_ d { f | E. y e. _pred ( x , A , R ) ta' } with typecode |-
26 9 25 nfcxfr ⊢ Ⅎ _ d H
27 26 nfuni ⊢ Ⅎ _ d ⋃ H
28 10 27 nfcxfr ⊢ Ⅎ _ d P
29 nfcv ⊢ Ⅎ _ d x
30 nfcv ⊢ Ⅎ _ d G
31 28 13 nfres ⊢ Ⅎ _ d P ↾ pred x A R
32 29 31 nfop ⊢ Ⅎ _ d x P ↾ pred x A R
33 11 32 nfcxfr ⊢ Ⅎ _ d Z
34 30 33 nffv ⊢ Ⅎ _ d G ⁡ Z
35 29 34 nfop ⊢ Ⅎ _ d x G ⁡ Z
36 35 nfsn ⊢ Ⅎ _ d x G ⁡ Z
37 28 36 nfun ⊢ Ⅎ _ d P ∪ x G ⁡ Z
38 12 37 nfcxfr ⊢ Ⅎ _ d Q
39 38 nfcrii ⊢ w ∈ Q → ∀ d w ∈ Q