Theorem fvmptd 5758

Description: Deduction version of fvmpt 5754. (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 5672	. 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 3185	. . . 4 |- ( ph -> [_ A / x ]_ B = C )
6		fvmptd.4	. . . 4 |- ( ph -> C e. V )
7	5, 6	eqeltrd 2309	. . 3 |- ( ph -> [_ A / x ]_ B e. V )
8		eqid 2232	. . . 4 |- ( x e. D |-> B ) = ( x e. D |-> B )
9	8	fvmpts 5755	. . 3 |- ( ( A e. D /\ [_ A / x ]_ B e. V ) -> ( ( x e. D |-> B ) ` A ) = [_ A / x ]_ B )
10	3, 7, 9	syl2anc 411	. 2 |- ( ph -> ( ( x e. D |-> B ) ` A ) = [_ A / x ]_ B )
11	2, 10, 5	3eqtrd 2269	1 |- ( ph -> ( F ` A ) = C )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   [_csb 3138   |-> cmpt 4171   ` cfv 5352

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-csb 3139  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-iota 5312  df-fun 5354  df-fv 5360

This theorem is referenced by:  fvmptd2  5759  fvmptdv2  5767  rdgivallem  6612  1stinl  7365  2ndinl  7366  1stinr  7367  2ndinr  7368  updjudhcoinlf  7371  updjudhcoinrg  7372  cardcl  7477  caucvgsrlemfv  8106  caucvgsrlemoffval  8111  axcaucvglemval  8212  negiso  9229  infrenegsupex  9926  iseqf1olemfvp  10872  seq3f1olemqsum  10875  ccatval1  11285  ccatval2  11286  infxrnegsupex  11948  climcvg1nlem  12034  isumshft  12176  mulgnngsum  13844  sraval  14585  lmfval  15058  blfvalps  15250  cdivcncfap  15469  peano4nninf  16784  peano3nninf  16785  nninfsellemeq  16792  nninfsellemeqinf  16794