Theorem fvmptf 5560

Description: Value of a function given by an ordered-pair class abstraction. This version of fvmptg 5544 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 2723	. . 3 |- ( C e. V -> C e. _V )
2		fvmptf.1	. . . 4 |- F/_ x A
3		fvmptf.2	. . . . . 6 |- F/_ x C
4	3	nfel1 2310	. . . . 5 |- F/ x C e. _V
5		fvmptf.4	. . . . . . . 8 |- F = ( x e. D |-> B )
6		nfmpt1 4057	. . . . . . . 8 |- F/_ x ( x e. D |-> B )
7	5, 6	nfcxfr 2296	. . . . . . 7 |- F/_ x F
8	7, 2	nffv 5478	. . . . . 6 |- F/_ x ( F ` A )
9	8, 3	nfeq 2307	. . . . 5 |- F/ x ( F ` A ) = C
10	4, 9	nfim 1552	. . . 4 |- F/ x ( C e. _V -> ( F ` A ) = C )
11		fvmptf.3	. . . . . 6 |- ( x = A -> B = C )
12	11	eleq1d 2226	. . . . 5 |- ( x = A -> ( B e. _V <-> C e. _V ) )
13		fveq2 5468	. . . . . 6 |- ( x = A -> ( F ` x ) = ( F ` A ) )
14	13, 11	eqeq12d 2172	. . . . 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 5551	. . . . 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 2777	. . 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 1335   e. wcel 2128   F/_wnfc 2286   _Vcvv 2712   |-> cmpt 4025   ` cfv 5170

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-csb 3032  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-iota 5135  df-fun 5172  df-fv 5178

This theorem is referenced by:  fvmptd3  5561  elfvmptrab1  5562  sumrbdclem  11274  fsum3  11284  isumss  11288  prodrbdclem  11468  prodmodclem2a  11473  zproddc  11476  fprodntrivap  11481  prodssdc  11486