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