Theorem fvmptf 5610

Description: Value of a function given by an ordered-pair class abstraction. This version of fvmptg 5594 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 2750	. . 3 |- ( C e. V -> C e. _V )
2		fvmptf.1	. . . 4 |- F/_ x A
3		fvmptf.2	. . . . . 6 |- F/_ x C
4	3	nfel1 2330	. . . . 5 |- F/ x C e. _V
5		fvmptf.4	. . . . . . . 8 |- F = ( x e. D |-> B )
6		nfmpt1 4098	. . . . . . . 8 |- F/_ x ( x e. D |-> B )
7	5, 6	nfcxfr 2316	. . . . . . 7 |- F/_ x F
8	7, 2	nffv 5527	. . . . . 6 |- F/_ x ( F ` A )
9	8, 3	nfeq 2327	. . . . 5 |- F/ x ( F ` A ) = C
10	4, 9	nfim 1572	. . . 4 |- F/ x ( C e. _V -> ( F ` A ) = C )
11		fvmptf.3	. . . . . 6 |- ( x = A -> B = C )
12	11	eleq1d 2246	. . . . 5 |- ( x = A -> ( B e. _V <-> C e. _V ) )
13		fveq2 5517	. . . . . 6 |- ( x = A -> ( F ` x ) = ( F ` A ) )
14	13, 11	eqeq12d 2192	. . . . 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 5601	. . . . 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 2804	. . 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 1353   e. wcel 2148   F/_wnfc 2306   _Vcvv 2739   |-> cmpt 4066   ` cfv 5218

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226

This theorem is referenced by:  fvmptd3  5611  elfvmptrab1  5612  sumrbdclem  11387  fsum3  11397  isumss  11401  prodrbdclem  11581  prodmodclem2a  11586  zproddc  11589  fprodntrivap  11594  prodssdc  11599  pcmpt  12343