Theorem fvmptd 5660

Description: Deduction version of fvmpt 5656. (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 5578	. 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 3140	. . . 4 |- ( ph -> [_ A / x ]_ B = C )
6		fvmptd.4	. . . 4 |- ( ph -> C e. V )
7	5, 6	eqeltrd 2282	. . 3 |- ( ph -> [_ A / x ]_ B e. V )
8		eqid 2205	. . . 4 |- ( x e. D |-> B ) = ( x e. D |-> B )
9	8	fvmpts 5657	. . 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 2242	1 |- ( ph -> ( F ` A ) = C )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   [_csb 3093   |-> cmpt 4105   ` cfv 5271

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-csb 3094  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-iota 5232  df-fun 5273  df-fv 5279

This theorem is referenced by:  fvmptd2  5661  fvmptdv2  5669  rdgivallem  6467  1stinl  7176  2ndinl  7177  1stinr  7178  2ndinr  7179  updjudhcoinlf  7182  updjudhcoinrg  7183  cardcl  7288  caucvgsrlemfv  7904  caucvgsrlemoffval  7909  axcaucvglemval  8010  negiso  9028  infrenegsupex  9715  iseqf1olemfvp  10655  seq3f1olemqsum  10658  ccatval1  11053  ccatval2  11054  infxrnegsupex  11574  climcvg1nlem  11660  isumshft  11801  mulgnngsum  13463  sraval  14199  lmfval  14664  blfvalps  14857  cdivcncfap  15076  peano4nninf  15943  peano3nninf  15944  nninfsellemeq  15951  nninfsellemeqinf  15953