Theorem fvmptf 6301

Description: Value of a function given by an ordered-pair class abstraction. This version of fvmptg 6280 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 3212	. . 3 |- ( C e. V -> C e. _V )
2		fvmptf.1	. . . 4 |- F/_ x A
3		fvmptf.2	. . . . . 6 |- F/_ x C
4	3	nfel1 2779	. . . . 5 |- F/ x C e. _V
5		fvmptf.4	. . . . . . . 8 |- F = ( x e. D |-> B )
6		nfmpt1 4747	. . . . . . . 8 |- F/_ x ( x e. D |-> B )
7	5, 6	nfcxfr 2762	. . . . . . 7 |- F/_ x F
8	7, 2	nffv 6198	. . . . . 6 |- F/_ x ( F ` A )
9	8, 3	nfeq 2776	. . . . 5 |- F/ x ( F ` A ) = C
10	4, 9	nfim 1825	. . . 4 |- F/ x ( C e. _V -> ( F ` A ) = C )
11		fvmptf.3	. . . . . 6 |- ( x = A -> B = C )
12	11	eleq1d 2686	. . . . 5 |- ( x = A -> ( B e. _V <-> C e. _V ) )
13		fveq2 6191	. . . . . 6 |- ( x = A -> ( F ` x ) = ( F ` A ) )
14	13, 11	eqeq12d 2637	. . . . 5 |- ( x = A -> ( ( F ` x ) = B <-> ( F ` A ) = C ) )
15	12, 14	imbi12d 334	. . . 4 |- ( x = A -> ( ( B e. _V -> ( F ` x ) = B ) <-> ( C e. _V -> ( F ` A ) = C ) ) )
16	5	fvmpt2 6291	. . . . 5 |- ( ( x e. D /\ B e. _V ) -> ( F ` x ) = B )
17	16	ex 450	. . . 4 |- ( x e. D -> ( B e. _V -> ( F ` x ) = B ) )
18	2, 10, 15, 17	vtoclgaf 3271	. . 3 |- ( A e. D -> ( C e. _V -> ( F ` A ) = C ) )
19	1, 18	syl5 34	. 2 |- ( A e. D -> ( C e. V -> ( F ` A ) = C ) )
20	19	imp 445	1 |- ( ( A e. D /\ C e. V ) -> ( F ` A ) = C )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   F/_wnfc 2751   _Vcvv 3200   |-> cmpt 4729   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896

This theorem is referenced by:  fvmptnf  6302  elfvmptrab1  6305  elovmpt3rab1  6893  rdgsucmptf  7524  frsucmpt  7533  fprodntriv  14672  prodss  14677  fprodefsum  14825  dvfsumabs  23786  dvfsumlem1  23789  dvfsumlem4  23792  dvfsum2  23797  dchrisumlem2  25179  dchrisumlem3  25180  ptrest  33408  hlhilset  37226  fvmptd3  39447  fsumsermpt  39811  mulc1cncfg  39821  expcnfg  39823  climsubmpt  39892  climeldmeqmpt  39900  climfveqmpt  39903  fnlimfvre  39906  fnlimfvre2  39909  climfveqmpt3  39914  climeldmeqmpt3  39921  climinf2mpt  39946  climinfmpt  39947  stoweidlem23  40240  stoweidlem34  40251  stoweidlem36  40253  wallispilem5  40286  stirlinglem4  40294  stirlinglem11  40301  stirlinglem12  40302  stirlinglem13  40303  stirlinglem14  40304  sge0lempt  40627  sge0isummpt2  40649  meadjiun  40683  hoimbl2  40879  vonhoire  40886