Theorem fvmptd 5703

Description: Deduction version of fvmpt 5701. (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 5331	. 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 3179	. . . 4 |- (ph -> [_A / x]_B = C)
6		fvmptd.4	. . . 4 |- (ph -> C e. V)
7	5, 6	eqeltrd 2427	. . 3 |- (ph -> [_A / x]_B e. V)
8		eqid 2353	. . . 4 |- (x e. D |-> B) = (x e. D |-> B)
9	8	fvmpts 5702	. . 3 |- ((A e. D /\ [_A / x]_B e. V) -> ((x e. D |-> B)` A) = [_A / x]_B)
10	3, 7, 9	syl2anc 642	. 2 |- (ph -> ((x e. D |-> B)` A) = [_A / x]_B)
11	2, 10, 5	3eqtrd 2389	1 |- (ph -> (F` A) = C)

Syntax hints:   -> wi 4   /\ wa 358   = wceq 1642   e. wcel 1710  [_csb 3137  ` cfv 4782   |-> cmpt 5652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-csb 3138  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-co 4727  df-ima 4728  df-id 4768  df-cnv 4786  df-rn 4787  df-dm 4788  df-fun 4790  df-fv 4796  df-mpt 5653

This theorem is referenced by: (None)