Theorem fvmptd 5723

Description: Deduction version of fvmpt 5719. (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 5637	. 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 3172	. . . 4 |- ( ph -> [_ A / x ]_ B = C )
6		fvmptd.4	. . . 4 |- ( ph -> C e. V )
7	5, 6	eqeltrd 2306	. . 3 |- ( ph -> [_ A / x ]_ B e. V )
8		eqid 2229	. . . 4 |- ( x e. D |-> B ) = ( x e. D |-> B )
9	8	fvmpts 5720	. . 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 2266	1 |- ( ph -> ( F ` A ) = C )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   [_csb 3125   |-> cmpt 4148   ` cfv 5324

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-sbc 3030  df-csb 3126  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-iota 5284  df-fun 5326  df-fv 5332

This theorem is referenced by:  fvmptd2  5724  fvmptdv2  5732  rdgivallem  6542  1stinl  7264  2ndinl  7265  1stinr  7266  2ndinr  7267  updjudhcoinlf  7270  updjudhcoinrg  7271  cardcl  7376  caucvgsrlemfv  8001  caucvgsrlemoffval  8006  axcaucvglemval  8107  negiso  9125  infrenegsupex  9818  iseqf1olemfvp  10762  seq3f1olemqsum  10765  ccatval1  11164  ccatval2  11165  infxrnegsupex  11814  climcvg1nlem  11900  isumshft  12041  mulgnngsum  13704  sraval  14441  lmfval  14907  blfvalps  15099  cdivcncfap  15318  peano4nninf  16544  peano3nninf  16545  nninfsellemeq  16552  nninfsellemeqinf  16554