Theorem fvmptd 5780

Description: Deduction version of fvmpt 5776. (Contributed by Scott Fenton, 18-Feb-2013.) (Revised by Mario Carneiro, 31-Aug-2015.)

Hypotheses

Ref	Expression
fvmptd.1	|- ( ph -> F = ( x e. D |-> B ) )
fvmptd.2	|- ( ( ph /\ x = A ) -> B = C )
fvmptd.3	|- ( ph -> A e. D )
fvmptd.4	|- ( ph -> C e. V )

Assertion

Ref	Expression
fvmptd	|- ( ph -> ( F ` A ) = C )

Distinct variable groups:   x, A   x, C   x, D   ph, x

Allowed substitution hints:   B( x)   F( x)   V( x)

Proof of Theorem fvmptd

Step	Hyp	Ref	Expression
1		fvmptd.1	. . 3 |- ( ph -> F = ( x e. D |-> B ) )
2	1	fveq1d 5692	. 2 |- ( ph -> ( F ` A ) = ( ( x e. D |-> B ) ` A ) )
3		fvmptd.3	. . 3 |- ( ph -> A e. D )
4		fvmptd.2	. . . . 5 |- ( ( ph /\ x = A ) -> B = C )
5	3, 4	csbied 3194	. . . 4 |- ( ph -> [_ A / x ]_ B = C )
6		fvmptd.4	. . . 4 |- ( ph -> C e. V )
7	5, 6	eqeltrd 2315	. . 3 |- ( ph -> [_ A / x ]_ B e. V )
8		eqid 2238	. . . 4 |- ( x e. D |-> B ) = ( x e. D |-> B )
9	8	fvmpts 5777	. . 3 |- ( ( A e. D /\ [_ A / x ]_ B e. V ) -> ( ( x e. D |-> B ) ` A ) = [_ A / x ]_ B )
10	3, 7, 9	syl2anc 415	. 2 |- ( ph -> ( ( x e. D |-> B ) ` A ) = [_ A / x ]_ B )
11	2, 10, 5	3eqtrd 2275	1 |- ( ph -> ( F ` A ) = C )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   [_csb 3147   |-> 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:  fvmptd2  5781  fvmptdv2  5789  rdgivallem  6642  1stinl  7404  2ndinl  7405  1stinr  7406  2ndinr  7407  updjudhcoinlf  7410  updjudhcoinrg  7411  cardcl  7516  caucvgsrlemfv  8148  caucvgsrlemoffval  8153  axcaucvglemval  8254  negiso  9275  infrenegsupex  9973  iseqf1olemfvp  10925  seq3f1olemqsum  10928  ccatval1  11343  ccatval2  11344  infxrnegsupex  12007  climcvg1nlem  12093  isumshft  12235  ballotfilemfval  13207  ballotfilemsv  13231  sraval  14746  lmfval  15217  blfvalps  15409  cdivcncfap  15628  peano4nninf  16954  peano3nninf  16955  nninfsellemeq  16962  nninfsellemeqinf  16964