Metamath Proof Explorer


Theorem bnj938

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 bnj938.1 D = ω
bnj938.2 No typesetting found for |- ( ta <-> ( f Fn m /\ ph' /\ ps' ) ) with typecode |-
bnj938.3 σ m D n = suc m p m
bnj938.4 No typesetting found for |- ( ph' <-> ( f ` (/) ) = _pred ( X , A , R ) ) with typecode |-
bnj938.5 No typesetting found for |- ( ps' <-> A. i e. _om ( suc i e. m -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) ) with typecode |-
Assertion bnj938 R FrSe A X A τ σ y f p pred y A R V

Proof

Step Hyp Ref Expression
1 bnj938.1 D = ω
2 bnj938.2 Could not format ( ta <-> ( f Fn m /\ ph' /\ ps' ) ) : No typesetting found for |- ( ta <-> ( f Fn m /\ ph' /\ ps' ) ) with typecode |-
3 bnj938.3 σ m D n = suc m p m
4 bnj938.4 Could not format ( ph' <-> ( f ` (/) ) = _pred ( X , A , R ) ) : No typesetting found for |- ( ph' <-> ( f ` (/) ) = _pred ( X , A , R ) ) with typecode |-
5 bnj938.5 Could not format ( ps' <-> A. i e. _om ( suc i e. m -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) ) : No typesetting found for |- ( ps' <-> A. i e. _om ( suc i e. m -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) ) with typecode |-
6 elisset X A x x = X
7 6 bnj706 R FrSe A X A τ σ x x = X
8 bnj291 R FrSe A X A τ σ R FrSe A τ σ X A
9 8 simplbi R FrSe A X A τ σ R FrSe A τ σ
10 bnj602 x = X pred x A R = pred X A R
11 10 eqeq2d x = X f = pred x A R f = pred X A R
12 11 4 bitr4di Could not format ( x = X -> ( ( f ` (/) ) = _pred ( x , A , R ) <-> ph' ) ) : No typesetting found for |- ( x = X -> ( ( f ` (/) ) = _pred ( x , A , R ) <-> ph' ) ) with typecode |-
13 12 3anbi2d Could not format ( x = X -> ( ( f Fn m /\ ( f ` (/) ) = _pred ( x , A , R ) /\ ps' ) <-> ( f Fn m /\ ph' /\ ps' ) ) ) : No typesetting found for |- ( x = X -> ( ( f Fn m /\ ( f ` (/) ) = _pred ( x , A , R ) /\ ps' ) <-> ( f Fn m /\ ph' /\ ps' ) ) ) with typecode |-
14 13 2 bitr4di Could not format ( x = X -> ( ( f Fn m /\ ( f ` (/) ) = _pred ( x , A , R ) /\ ps' ) <-> ta ) ) : No typesetting found for |- ( x = X -> ( ( f Fn m /\ ( f ` (/) ) = _pred ( x , A , R ) /\ ps' ) <-> ta ) ) with typecode |-
15 14 3anbi2d Could not format ( x = X -> ( ( R _FrSe A /\ ( f Fn m /\ ( f ` (/) ) = _pred ( x , A , R ) /\ ps' ) /\ si ) <-> ( R _FrSe A /\ ta /\ si ) ) ) : No typesetting found for |- ( x = X -> ( ( R _FrSe A /\ ( f Fn m /\ ( f ` (/) ) = _pred ( x , A , R ) /\ ps' ) /\ si ) <-> ( R _FrSe A /\ ta /\ si ) ) ) with typecode |-
16 9 15 syl5ibr Could not format ( x = X -> ( ( R _FrSe A /\ X e. A /\ ta /\ si ) -> ( R _FrSe A /\ ( f Fn m /\ ( f ` (/) ) = _pred ( x , A , R ) /\ ps' ) /\ si ) ) ) : No typesetting found for |- ( x = X -> ( ( R _FrSe A /\ X e. A /\ ta /\ si ) -> ( R _FrSe A /\ ( f Fn m /\ ( f ` (/) ) = _pred ( x , A , R ) /\ ps' ) /\ si ) ) ) with typecode |-
17 biid Could not format ( ( f Fn m /\ ( f ` (/) ) = _pred ( x , A , R ) /\ ps' ) <-> ( f Fn m /\ ( f ` (/) ) = _pred ( x , A , R ) /\ ps' ) ) : No typesetting found for |- ( ( f Fn m /\ ( f ` (/) ) = _pred ( x , A , R ) /\ ps' ) <-> ( f Fn m /\ ( f ` (/) ) = _pred ( x , A , R ) /\ ps' ) ) with typecode |-
18 biid f = pred x A R f = pred x A R
19 1 17 3 18 5 bnj546 Could not format ( ( R _FrSe A /\ ( f Fn m /\ ( f ` (/) ) = _pred ( x , A , R ) /\ ps' ) /\ si ) -> U_ y e. ( f ` p ) _pred ( y , A , R ) e. _V ) : No typesetting found for |- ( ( R _FrSe A /\ ( f Fn m /\ ( f ` (/) ) = _pred ( x , A , R ) /\ ps' ) /\ si ) -> U_ y e. ( f ` p ) _pred ( y , A , R ) e. _V ) with typecode |-
20 16 19 syl6 x = X R FrSe A X A τ σ y f p pred y A R V
21 20 exlimiv x x = X R FrSe A X A τ σ y f p pred y A R V
22 7 21 mpcom R FrSe A X A τ σ y f p pred y A R V