Theorem oemapval 8580

Description: Value of the relation T. (Contributed by Mario Carneiro, 28-May-2015.)

Hypotheses

Ref	Expression
cantnfs.s	|- S = dom ( A CNF B )
cantnfs.a	|- ( ph -> A e. On )
cantnfs.b	|- ( ph -> B e. On )
oemapval.t	|- T = { <. x , y >. | E. z e. B ( ( x ` z ) e. ( y ` z ) /\ A. w e. B ( z e. w -> ( x ` w ) = ( y ` w ) ) ) }
oemapval.f	|- ( ph -> F e. S )
oemapval.g	|- ( ph -> G e. S )

Assertion

Ref	Expression
oemapval	|- ( ph -> ( F T G <-> E. z e. B ( ( F ` z ) e. ( G ` z ) /\ A. w e. B ( z e. w -> ( F ` w ) = ( G ` w ) ) ) ) )

Distinct variable groups:   x, w, y, z, B   w, A, x, y, z   w, F, x, y, z   x, S, y, z   w, G, x, y, z   ph, x, y, z

Allowed substitution hints:   ph( w)   S( w)   T( x, y, z, w)

Proof of Theorem oemapval

Step	Hyp	Ref	Expression
1		oemapval.f	. 2 |- ( ph -> F e. S )
2		oemapval.g	. 2 |- ( ph -> G e. S )
3		fveq1 6190	. . . . . 6 |- ( x = F -> ( x ` z ) = ( F ` z ) )
4		fveq1 6190	. . . . . 6 |- ( y = G -> ( y ` z ) = ( G ` z ) )
5		eleq12 2691	. . . . . 6 |- ( ( ( x ` z ) = ( F ` z ) /\ ( y ` z ) = ( G ` z ) ) -> ( ( x ` z ) e. ( y ` z ) <-> ( F ` z ) e. ( G ` z ) ) )
6	3, 4, 5	syl2an 494	. . . . 5 |- ( ( x = F /\ y = G ) -> ( ( x ` z ) e. ( y ` z ) <-> ( F ` z ) e. ( G ` z ) ) )
7		fveq1 6190	. . . . . . . 8 |- ( x = F -> ( x ` w ) = ( F ` w ) )
8		fveq1 6190	. . . . . . . 8 |- ( y = G -> ( y ` w ) = ( G ` w ) )
9	7, 8	eqeqan12d 2638	. . . . . . 7 |- ( ( x = F /\ y = G ) -> ( ( x ` w ) = ( y ` w ) <-> ( F ` w ) = ( G ` w ) ) )
10	9	imbi2d 330	. . . . . 6 |- ( ( x = F /\ y = G ) -> ( ( z e. w -> ( x ` w ) = ( y ` w ) ) <-> ( z e. w -> ( F ` w ) = ( G ` w ) ) ) )
11	10	ralbidv 2986	. . . . 5 |- ( ( x = F /\ y = G ) -> ( A. w e. B ( z e. w -> ( x ` w ) = ( y ` w ) ) <-> A. w e. B ( z e. w -> ( F ` w ) = ( G ` w ) ) ) )
12	6, 11	anbi12d 747	. . . 4 |- ( ( x = F /\ y = G ) -> ( ( ( x ` z ) e. ( y ` z ) /\ A. w e. B ( z e. w -> ( x ` w ) = ( y ` w ) ) ) <-> ( ( F ` z ) e. ( G ` z ) /\ A. w e. B ( z e. w -> ( F ` w ) = ( G ` w ) ) ) ) )
13	12	rexbidv 3052	. . 3 |- ( ( x = F /\ y = G ) -> ( E. z e. B ( ( x ` z ) e. ( y ` z ) /\ A. w e. B ( z e. w -> ( x ` w ) = ( y ` w ) ) ) <-> E. z e. B ( ( F ` z ) e. ( G ` z ) /\ A. w e. B ( z e. w -> ( F ` w ) = ( G ` w ) ) ) ) )
14		oemapval.t	. . 3 |- T = { <. x , y >. | E. z e. B ( ( x ` z ) e. ( y ` z ) /\ A. w e. B ( z e. w -> ( x ` w ) = ( y ` w ) ) ) }
15	13, 14	brabga 4989	. 2 |- ( ( F e. S /\ G e. S ) -> ( F T G <-> E. z e. B ( ( F ` z ) e. ( G ` z ) /\ A. w e. B ( z e. w -> ( F ` w ) = ( G ` w ) ) ) ) )
16	1, 2, 15	syl2anc 693	1 |- ( ph -> ( F T G <-> E. z e. B ( ( F ` z ) e. ( G ` z ) /\ A. w e. B ( z e. w -> ( F ` w ) = ( G ` w ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   class class class wbr 4653   {copab 4712   dom cdm 5114   Oncon0 5723   ` cfv 5888  (class class class)co 6650   CNF ccnf 8558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-iota 5851  df-fv 5896

This theorem is referenced by:  oemapvali  8581