Theorem fvmptf 5578

Description: Value of a function given by an ordered-pair class abstraction. This version of fvmptg 5562 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 2737	. . 3 |- ( C e. V -> C e. _V )
2		fvmptf.1	. . . 4 |- F/_ x A
3		fvmptf.2	. . . . . 6 |- F/_ x C
4	3	nfel1 2319	. . . . 5 |- F/ x C e. _V
5		fvmptf.4	. . . . . . . 8 |- F = ( x e. D |-> B )
6		nfmpt1 4075	. . . . . . . 8 |- F/_ x ( x e. D |-> B )
7	5, 6	nfcxfr 2305	. . . . . . 7 |- F/_ x F
8	7, 2	nffv 5496	. . . . . 6 |- F/_ x ( F ` A )
9	8, 3	nfeq 2316	. . . . 5 |- F/ x ( F ` A ) = C
10	4, 9	nfim 1560	. . . 4 |- F/ x ( C e. _V -> ( F ` A ) = C )
11		fvmptf.3	. . . . . 6 |- ( x = A -> B = C )
12	11	eleq1d 2235	. . . . 5 |- ( x = A -> ( B e. _V <-> C e. _V ) )
13		fveq2 5486	. . . . . 6 |- ( x = A -> ( F ` x ) = ( F ` A ) )
14	13, 11	eqeq12d 2180	. . . . 5 |- ( x = A -> ( ( F ` x ) = B <-> ( F ` A ) = C ) )
15	12, 14	imbi12d 233	. . . 4 |- ( x = A -> ( ( B e. _V -> ( F ` x ) = B ) <-> ( C e. _V -> ( F ` A ) = C ) ) )
16	5	fvmpt2 5569	. . . . 5 |- ( ( x e. D /\ B e. _V ) -> ( F ` x ) = B )
17	16	ex 114	. . . 4 |- ( x e. D -> ( B e. _V -> ( F ` x ) = B ) )
18	2, 10, 15, 17	vtoclgaf 2791	. . 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 123	1 |- ( ( A e. D /\ C e. V ) -> ( F ` A ) = C )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   F/_wnfc 2295   _Vcvv 2726   |-> cmpt 4043   ` cfv 5188

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196

This theorem is referenced by:  fvmptd3  5579  elfvmptrab1  5580  sumrbdclem  11318  fsum3  11328  isumss  11332  prodrbdclem  11512  prodmodclem2a  11517  zproddc  11520  fprodntrivap  11525  prodssdc  11530  pcmpt  12273