Theorem fvmptf 5792

Description: Value of a function given by an ordered-pair class abstraction. This version of fvmptg 5775 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 2833	. . 3 |- ( C e. V -> C e. _V )
2		fvmptf.1	. . . 4 |- F/_ x A
3		fvmptf.2	. . . . . 6 |- F/_ x C
4	3	nfel1 2403	. . . . 5 |- F/ x C e. _V
5		fvmptf.4	. . . . . . . 8 |- F = ( x e. D |-> B )
6		nfmpt1 4219	. . . . . . . 8 |- F/_ x ( x e. D |-> B )
7	5, 6	nfcxfr 2389	. . . . . . 7 |- F/_ x F
8	7, 2	nffv 5700	. . . . . 6 |- F/_ x ( F ` A )
9	8, 3	nfeq 2400	. . . . 5 |- F/ x ( F ` A ) = C
10	4, 9	nfim 1625	. . . 4 |- F/ x ( C e. _V -> ( F ` A ) = C )
11		fvmptf.3	. . . . . 6 |- ( x = A -> B = C )
12	11	eleq1d 2307	. . . . 5 |- ( x = A -> ( B e. _V <-> C e. _V ) )
13		fveq2 5690	. . . . . 6 |- ( x = A -> ( F ` x ) = ( F ` A ) )
14	13, 11	eqeq12d 2253	. . . . 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 5783	. . . . 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 2888	. . 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 1402   e. wcel 2209   F/_wnfc 2379   _Vcvv 2821   |-> cmpt 4187   ` cfv 5372

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fun 5374  df-fv 5380

This theorem is referenced by:  fvmptd3  5793  elfvmptrab1  5794  sumrbdclem  12122  fsum3  12132  isumss  12136  prodrbdclem  12316  prodmodclem2a  12321  zproddc  12324  fprodntrivap  12329  prodssdc  12334  pcmpt  13100  gsummptfidmadd  14138