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Metamath Proof Explorer

Theorem bnj1296

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 bnj1296.1 
|- B = { d | ( d C_ A /\ A. x e. d _pred ( x , A , R ) C_ d ) }
bnj1296.2 
|- Y = <. x , ( f |` _pred ( x , A , R ) ) >.
bnj1296.3 
|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
bnj1296.4 
|- D = ( dom g i^i dom h )
bnj1296.5 
|- E = { x e. D | ( g ` x ) =/= ( h ` x ) }
bnj1296.6 
|- ( ph <-> ( R _FrSe A /\ g e. C /\ h e. C /\ ( g |` D ) =/= ( h |` D ) ) )
bnj1296.7 
|- ( ps <-> ( ph /\ x e. E /\ A. y e. E -. y R x ) )
bnj1296.18 
|- ( ps -> ( g |` _pred ( x , A , R ) ) = ( h |` _pred ( x , A , R ) ) )
bnj1296.9 
|- Z = <. x , ( g |` _pred ( x , A , R ) ) >.
bnj1296.10 
|- K = { g | E. d e. B ( g Fn d /\ A. x e. d ( g ` x ) = ( G ` Z ) ) }
bnj1296.11 
|- W = <. x , ( h |` _pred ( x , A , R ) ) >.
bnj1296.12 
|- L = { h | E. d e. B ( h Fn d /\ A. x e. d ( h ` x ) = ( G ` W ) ) }
Assertion bnj1296 
|- ( ps -> ( g ` x ) = ( h ` x ) )

Proof

Step Hyp Ref Expression
1 bnj1296.1 
 |-  B = { d | ( d C_ A /\ A. x e. d _pred ( x , A , R ) C_ d ) }
2 bnj1296.2 
 |-  Y = <. x , ( f |` _pred ( x , A , R ) ) >.
3 bnj1296.3 
 |-  C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
4 bnj1296.4 
 |-  D = ( dom g i^i dom h )
5 bnj1296.5 
 |-  E = { x e. D | ( g ` x ) =/= ( h ` x ) }
6 bnj1296.6 
 |-  ( ph <-> ( R _FrSe A /\ g e. C /\ h e. C /\ ( g |` D ) =/= ( h |` D ) ) )
7 bnj1296.7 
 |-  ( ps <-> ( ph /\ x e. E /\ A. y e. E -. y R x ) )
8 bnj1296.18 
 |-  ( ps -> ( g |` _pred ( x , A , R ) ) = ( h |` _pred ( x , A , R ) ) )
9 bnj1296.9 
 |-  Z = <. x , ( g |` _pred ( x , A , R ) ) >.
10 bnj1296.10 
 |-  K = { g | E. d e. B ( g Fn d /\ A. x e. d ( g ` x ) = ( G ` Z ) ) }
11 bnj1296.11 
 |-  W = <. x , ( h |` _pred ( x , A , R ) ) >.
12 bnj1296.12 
 |-  L = { h | E. d e. B ( h Fn d /\ A. x e. d ( h ` x ) = ( G ` W ) ) }
13 8 opeq2d 
 |-  ( ps -> <. x , ( g |` _pred ( x , A , R ) ) >. = <. x , ( h |` _pred ( x , A , R ) ) >. )
14 13 9 11 3eqtr4g 
 |-  ( ps -> Z = W )
15 14 fveq2d 
 |-  ( ps -> ( G ` Z ) = ( G ` W ) )
16 10 bnj1436 
 |-  ( g e. K -> E. d e. B ( g Fn d /\ A. x e. d ( g ` x ) = ( G ` Z ) ) )
17 fndm 
 |-  ( g Fn d -> dom g = d )
18 17 anim1i 
 |-  ( ( g Fn d /\ A. x e. d ( g ` x ) = ( G ` Z ) ) -> ( dom g = d /\ A. x e. d ( g ` x ) = ( G ` Z ) ) )
19 16 18 bnj31 
 |-  ( g e. K -> E. d e. B ( dom g = d /\ A. x e. d ( g ` x ) = ( G ` Z ) ) )
20 raleq 
 |-  ( dom g = d -> ( A. x e. dom g ( g ` x ) = ( G ` Z ) <-> A. x e. d ( g ` x ) = ( G ` Z ) ) )
21 20 pm5.32i 
 |-  ( ( dom g = d /\ A. x e. dom g ( g ` x ) = ( G ` Z ) ) <-> ( dom g = d /\ A. x e. d ( g ` x ) = ( G ` Z ) ) )
22 21 rexbii 
 |-  ( E. d e. B ( dom g = d /\ A. x e. dom g ( g ` x ) = ( G ` Z ) ) <-> E. d e. B ( dom g = d /\ A. x e. d ( g ` x ) = ( G ` Z ) ) )
23 19 22 sylibr 
 |-  ( g e. K -> E. d e. B ( dom g = d /\ A. x e. dom g ( g ` x ) = ( G ` Z ) ) )
24 simpr 
 |-  ( ( dom g = d /\ A. x e. dom g ( g ` x ) = ( G ` Z ) ) -> A. x e. dom g ( g ` x ) = ( G ` Z ) )
25 23 24 bnj31 
 |-  ( g e. K -> E. d e. B A. x e. dom g ( g ` x ) = ( G ` Z ) )
26 25 bnj1265 
 |-  ( g e. K -> A. x e. dom g ( g ` x ) = ( G ` Z ) )
27 2 3 9 10 bnj1234 
 |-  C = K
28 26 27 eleq2s 
 |-  ( g e. C -> A. x e. dom g ( g ` x ) = ( G ` Z ) )
29 6 28 bnj770 
 |-  ( ph -> A. x e. dom g ( g ` x ) = ( G ` Z ) )
30 7 29 bnj835 
 |-  ( ps -> A. x e. dom g ( g ` x ) = ( G ` Z ) )
31 4 bnj1292 
 |-  D C_ dom g
32 5 7 bnj1212 
 |-  ( ps -> x e. D )
33 31 32 bnj1213 
 |-  ( ps -> x e. dom g )
34 30 33 bnj1294 
 |-  ( ps -> ( g ` x ) = ( G ` Z ) )
35 12 bnj1436 
 |-  ( h e. L -> E. d e. B ( h Fn d /\ A. x e. d ( h ` x ) = ( G ` W ) ) )
36 fndm 
 |-  ( h Fn d -> dom h = d )
37 36 anim1i 
 |-  ( ( h Fn d /\ A. x e. d ( h ` x ) = ( G ` W ) ) -> ( dom h = d /\ A. x e. d ( h ` x ) = ( G ` W ) ) )
38 35 37 bnj31 
 |-  ( h e. L -> E. d e. B ( dom h = d /\ A. x e. d ( h ` x ) = ( G ` W ) ) )
39 raleq 
 |-  ( dom h = d -> ( A. x e. dom h ( h ` x ) = ( G ` W ) <-> A. x e. d ( h ` x ) = ( G ` W ) ) )
40 39 pm5.32i 
 |-  ( ( dom h = d /\ A. x e. dom h ( h ` x ) = ( G ` W ) ) <-> ( dom h = d /\ A. x e. d ( h ` x ) = ( G ` W ) ) )
41 40 rexbii 
 |-  ( E. d e. B ( dom h = d /\ A. x e. dom h ( h ` x ) = ( G ` W ) ) <-> E. d e. B ( dom h = d /\ A. x e. d ( h ` x ) = ( G ` W ) ) )
42 38 41 sylibr 
 |-  ( h e. L -> E. d e. B ( dom h = d /\ A. x e. dom h ( h ` x ) = ( G ` W ) ) )
43 simpr 
 |-  ( ( dom h = d /\ A. x e. dom h ( h ` x ) = ( G ` W ) ) -> A. x e. dom h ( h ` x ) = ( G ` W ) )
44 42 43 bnj31 
 |-  ( h e. L -> E. d e. B A. x e. dom h ( h ` x ) = ( G ` W ) )
45 44 bnj1265 
 |-  ( h e. L -> A. x e. dom h ( h ` x ) = ( G ` W ) )
46 2 3 11 12 bnj1234 
 |-  C = L
47 45 46 eleq2s 
 |-  ( h e. C -> A. x e. dom h ( h ` x ) = ( G ` W ) )
48 6 47 bnj771 
 |-  ( ph -> A. x e. dom h ( h ` x ) = ( G ` W ) )
49 7 48 bnj835 
 |-  ( ps -> A. x e. dom h ( h ` x ) = ( G ` W ) )
50 4 bnj1293 
 |-  D C_ dom h
51 50 32 bnj1213 
 |-  ( ps -> x e. dom h )
52 49 51 bnj1294 
 |-  ( ps -> ( h ` x ) = ( G ` W ) )
53 15 34 52 3eqtr4d 
 |-  ( ps -> ( g ` x ) = ( h ` x ) )