Theorem fvmptf 5671

Description: Value of a function given by an ordered-pair class abstraction. This version of fvmptg 5654 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 8-Nov-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypotheses

Ref	Expression
fvmptf.1	|- F/_ x A
fvmptf.2	|- F/_ x C
fvmptf.3	|- ( x = A -> B = C )
fvmptf.4	|- F = ( x e. D |-> B )

Assertion

Ref	Expression
fvmptf	|- ( ( A e. D /\ C e. V ) -> ( F ` A ) = C )

Distinct variable group:   x, D

Allowed substitution hints:   A( x)   B( x)   C( x)   F( x)   V( x)

Proof of Theorem fvmptf

Step	Hyp	Ref	Expression
1		elex 2782	. . 3 |- ( C e. V -> C e. _V )
2		fvmptf.1	. . . 4 |- F/_ x A
3		fvmptf.2	. . . . . 6 |- F/_ x C
4	3	nfel1 2358	. . . . 5 |- F/ x C e. _V
5		fvmptf.4	. . . . . . . 8 |- F = ( x e. D |-> B )
6		nfmpt1 4136	. . . . . . . 8 |- F/_ x ( x e. D |-> B )
7	5, 6	nfcxfr 2344	. . . . . . 7 |- F/_ x F
8	7, 2	nffv 5585	. . . . . 6 |- F/_ x ( F ` A )
9	8, 3	nfeq 2355	. . . . 5 |- F/ x ( F ` A ) = C
10	4, 9	nfim 1594	. . . 4 |- F/ x ( C e. _V -> ( F ` A ) = C )
11		fvmptf.3	. . . . . 6 |- ( x = A -> B = C )
12	11	eleq1d 2273	. . . . 5 |- ( x = A -> ( B e. _V <-> C e. _V ) )
13		fveq2 5575	. . . . . 6 |- ( x = A -> ( F ` x ) = ( F ` A ) )
14	13, 11	eqeq12d 2219	. . . . 5 |- ( x = A -> ( ( F ` x ) = B <-> ( F ` A ) = C ) )
15	12, 14	imbi12d 234	. . . 4 |- ( x = A -> ( ( B e. _V -> ( F ` x ) = B ) <-> ( C e. _V -> ( F ` A ) = C ) ) )
16	5	fvmpt2 5662	. . . . 5 |- ( ( x e. D /\ B e. _V ) -> ( F ` x ) = B )
17	16	ex 115	. . . 4 |- ( x e. D -> ( B e. _V -> ( F ` x ) = B ) )
18	2, 10, 15, 17	vtoclgaf 2837	. . 3 |- ( A e. D -> ( C e. _V -> ( F ` A ) = C ) )
19	1, 18	syl5 32	. 2 |- ( A e. D -> ( C e. V -> ( F ` A ) = C ) )
20	19	imp 124	1 |- ( ( A e. D /\ C e. V ) -> ( F ` A ) = C )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1372   e. wcel 2175   F/_wnfc 2334   _Vcvv 2771   |-> cmpt 4104   ` cfv 5270

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-sbc 2998  df-csb 3093  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-iota 5231  df-fun 5272  df-fv 5278

This theorem is referenced by:  fvmptd3  5672  elfvmptrab1  5673  sumrbdclem  11659  fsum3  11669  isumss  11673  prodrbdclem  11853  prodmodclem2a  11858  zproddc  11861  fprodntrivap  11866  prodssdc  11871  pcmpt  12637