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Metamath Proof Explorer

Theorem bnj1030

Description: Technical lemma for bnj69 . 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 bnj1030.1 ⊢ φ ↔ f ⁡ ∅ = pred X A R
bnj1030.2 ⊢ ψ ↔ ∀ i ∈ ω suc ⁡ i ∈ n → f ⁡ suc ⁡ i = ⋃ y ∈ f ⁡ i pred y A R
bnj1030.3 ⊢ χ ↔ n ∈ D ∧ f Fn n ∧ φ ∧ ψ
bnj1030.4 ⊢ θ ↔ R FrSe A ∧ X ∈ A
bnj1030.5 ⊢ τ ↔ B ∈ V ∧ TrFo B A R ∧ pred X A R ⊆ B
bnj1030.6 ⊢ ζ ↔ i ∈ n ∧ z ∈ f ⁡ i
bnj1030.7 ⊢ D = ω ∖ ∅
bnj1030.8 ⊢ K = f | ∃ n ∈ D f Fn n ∧ φ ∧ ψ
bnj1030.9 ⊢ η ↔ f ∈ K ∧ i ∈ dom ⁡ f → f ⁡ i ⊆ B
bnj1030.10 ⊢ ρ ↔ ∀ j ∈ n j E i → [˙j / i]˙ η
bnj1030.11 No typesetting found for |- ( ph' <-> [. j / i ]. ph ) with typecode |-
bnj1030.12 No typesetting found for |- ( ps' <-> [. j / i ]. ps ) with typecode |-
bnj1030.13 No typesetting found for |- ( ch' <-> [. j / i ]. ch ) with typecode |-
bnj1030.14 No typesetting found for |- ( th' <-> [. j / i ]. th ) with typecode |-
bnj1030.15 No typesetting found for |- ( ta' <-> [. j / i ]. ta ) with typecode |-
bnj1030.16 No typesetting found for |- ( ze' <-> [. j / i ]. ze ) with typecode |-
bnj1030.17 No typesetting found for |- ( et' <-> [. j / i ]. et ) with typecode |-
bnj1030.18 No typesetting found for |- ( si <-> ( ( j e. n /\ j _E i ) -> et' ) ) with typecode |-
bnj1030.19 No typesetting found for |- ( ph0 <-> ( i e. n /\ si /\ f e. K /\ i e. dom f ) ) with typecode |-
Assertion bnj1030 ⊢ θ ∧ τ → trCl X A R ⊆ B

Proof

Step Hyp Ref Expression
1 bnj1030.1 ⊢ φ ↔ f ⁡ ∅ = pred X A R
2 bnj1030.2 ⊢ ψ ↔ ∀ i ∈ ω suc ⁡ i ∈ n → f ⁡ suc ⁡ i = ⋃ y ∈ f ⁡ i pred y A R
3 bnj1030.3 ⊢ χ ↔ n ∈ D ∧ f Fn n ∧ φ ∧ ψ
4 bnj1030.4 ⊢ θ ↔ R FrSe A ∧ X ∈ A
5 bnj1030.5 ⊢ τ ↔ B ∈ V ∧ TrFo B A R ∧ pred X A R ⊆ B
6 bnj1030.6 ⊢ ζ ↔ i ∈ n ∧ z ∈ f ⁡ i
7 bnj1030.7 ⊢ D = ω ∖ ∅
8 bnj1030.8 ⊢ K = f | ∃ n ∈ D f Fn n ∧ φ ∧ ψ
9 bnj1030.9 ⊢ η ↔ f ∈ K ∧ i ∈ dom ⁡ f → f ⁡ i ⊆ B
10 bnj1030.10 ⊢ ρ ↔ ∀ j ∈ n j E i → [˙j / i]˙ η
11 bnj1030.11 Could not format ( ph' <-> [. j / i ]. ph ) : No typesetting found for |- ( ph' <-> [. j / i ]. ph ) with typecode |-
12 bnj1030.12 Could not format ( ps' <-> [. j / i ]. ps ) : No typesetting found for |- ( ps' <-> [. j / i ]. ps ) with typecode |-
13 bnj1030.13 Could not format ( ch' <-> [. j / i ]. ch ) : No typesetting found for |- ( ch' <-> [. j / i ]. ch ) with typecode |-
14 bnj1030.14 Could not format ( th' <-> [. j / i ]. th ) : No typesetting found for |- ( th' <-> [. j / i ]. th ) with typecode |-
15 bnj1030.15 Could not format ( ta' <-> [. j / i ]. ta ) : No typesetting found for |- ( ta' <-> [. j / i ]. ta ) with typecode |-
16 bnj1030.16 Could not format ( ze' <-> [. j / i ]. ze ) : No typesetting found for |- ( ze' <-> [. j / i ]. ze ) with typecode |-
17 bnj1030.17 Could not format ( et' <-> [. j / i ]. et ) : No typesetting found for |- ( et' <-> [. j / i ]. et ) with typecode |-
18 bnj1030.18 Could not format ( si <-> ( ( j e. n /\ j _E i ) -> et' ) ) : No typesetting found for |- ( si <-> ( ( j e. n /\ j _E i ) -> et' ) ) with typecode |-
19 bnj1030.19 Could not format ( ph0 <-> ( i e. n /\ si /\ f e. K /\ i e. dom f ) ) : No typesetting found for |- ( ph0 <-> ( i e. n /\ si /\ f e. K /\ i e. dom f ) ) with typecode |-
20 19.23vv ⊢ ∀ n ∀ i θ ∧ τ ∧ χ ∧ ζ → z ∈ B ↔ ∃ n ∃ i θ ∧ τ ∧ χ ∧ ζ → z ∈ B
21 20 albii ⊢ ∀ f ∀ n ∀ i θ ∧ τ ∧ χ ∧ ζ → z ∈ B ↔ ∀ f ∃ n ∃ i θ ∧ τ ∧ χ ∧ ζ → z ∈ B
22 19.23v ⊢ ∀ f ∃ n ∃ i θ ∧ τ ∧ χ ∧ ζ → z ∈ B ↔ ∃ f ∃ n ∃ i θ ∧ τ ∧ χ ∧ ζ → z ∈ B
23 21 22 bitri ⊢ ∀ f ∀ n ∀ i θ ∧ τ ∧ χ ∧ ζ → z ∈ B ↔ ∃ f ∃ n ∃ i θ ∧ τ ∧ χ ∧ ζ → z ∈ B
24 7 bnj1071 ⊢ n ∈ D → E Fr n
25 3 24 bnj769 ⊢ χ → E Fr n
26 25 bnj707 ⊢ θ ∧ τ ∧ χ ∧ ζ → E Fr n
27 2 8 9 17 bnj1123 Could not format ( et' <-> ( ( f e. K /\ j e. dom f ) -> ( f ` j ) C_ B ) ) : No typesetting found for |- ( et' <-> ( ( f e. K /\ j e. dom f ) -> ( f ` j ) C_ B ) ) with typecode |-
28 2 3 5 7 18 19 27 bnj1118 Could not format E. j ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) -> ( f ` i ) C_ B ) : No typesetting found for |- E. j ( ( i =/= (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) -> ( f ` i ) C_ B ) with typecode |-
29 1 3 5 bnj1097 Could not format ( ( i = (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) -> ( f ` i ) C_ B ) : No typesetting found for |- ( ( i = (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) -> ( f ` i ) C_ B ) with typecode |-
30 28 29 bnj1109 Could not format E. j ( ( ( th /\ ta /\ ch ) /\ ph0 ) -> ( f ` i ) C_ B ) : No typesetting found for |- E. j ( ( ( th /\ ta /\ ch ) /\ ph0 ) -> ( f ` i ) C_ B ) with typecode |-
31 30 2 3 bnj1093 Could not format ( ( th /\ ta /\ ch /\ ze ) -> A. i E. j ( ph0 -> ( f ` i ) C_ B ) ) : No typesetting found for |- ( ( th /\ ta /\ ch /\ ze ) -> A. i E. j ( ph0 -> ( f ` i ) C_ B ) ) with typecode |-
32 9 10 17 18 19 31 bnj1090 ⊢ θ ∧ τ ∧ χ ∧ ζ → ∀ i ∈ n ρ → η
33 vex ⊢ n ∈ V
34 33 10 bnj110 ⊢ E Fr n ∧ ∀ i ∈ n ρ → η → ∀ i ∈ n η
35 26 32 34 syl2anc ⊢ θ ∧ τ ∧ χ ∧ ζ → ∀ i ∈ n η
36 4 5 3 6 9 35 8 bnj1121 ⊢ θ ∧ τ ∧ χ ∧ ζ → z ∈ B
37 36 gen2 ⊢ ∀ n ∀ i θ ∧ τ ∧ χ ∧ ζ → z ∈ B
38 23 37 mpgbi ⊢ ∃ f ∃ n ∃ i θ ∧ τ ∧ χ ∧ ζ → z ∈ B
39 1 2 3 4 5 6 7 8 38 bnj1034 ⊢ θ ∧ τ → trCl X A R ⊆ B