Theorem fvmptdf 5311

Description: Alternate deduction version of fvmpt 5302, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)

Hypotheses

Ref	Expression
fvmptdf.1	|- ( ph -> A e. D )
fvmptdf.2	|- ( ( ph /\ x = A ) -> B e. V )
fvmptdf.3	|- ( ( ph /\ x = A ) -> ( ( F ` A ) = B -> ps ) )
fvmptdf.4	|- F/_ x F
fvmptdf.5	|- F/ x ps

Assertion

Ref	Expression
fvmptdf	|- ( ph -> ( F = ( x e. D |-> B ) -> ps ) )

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

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

Proof of Theorem fvmptdf

Step	Hyp	Ref	Expression
1		nfv 1462	. 2 |- F/ x ph
2		fvmptdf.4	. . . 4 |- F/_ x F
3		nfmpt1 3892	. . . 4 |- F/_ x ( x e. D |-> B )
4	2, 3	nfeq 2230	. . 3 |- F/ x F = ( x e. D |-> B )
5		fvmptdf.5	. . 3 |- F/ x ps
6	4, 5	nfim 1505	. 2 |- F/ x ( F = ( x e. D |-> B ) -> ps )
7		fvmptdf.1	. . . 4 |- ( ph -> A e. D )
8		elex 2619	. . . 4 |- ( A e. D -> A e. _V )
9	7, 8	syl 14	. . 3 |- ( ph -> A e. _V )
10		isset 2614	. . 3 |- ( A e. _V <-> E. x x = A )
11	9, 10	sylib 120	. 2 |- ( ph -> E. x x = A )
12		fveq1 5229	. . 3 |- ( F = ( x e. D |-> B ) -> ( F ` A ) = ( ( x e. D |-> B ) ` A ) )
13		simpr 108	. . . . . . 7 |- ( ( ph /\ x = A ) -> x = A )
14	13	fveq2d 5234	. . . . . 6 |- ( ( ph /\ x = A ) -> ( ( x e. D |-> B ) ` x ) = ( ( x e. D |-> B ) ` A ) )
15	7	adantr 270	. . . . . . . 8 |- ( ( ph /\ x = A ) -> A e. D )
16	13, 15	eqeltrd 2159	. . . . . . 7 |- ( ( ph /\ x = A ) -> x e. D )
17		fvmptdf.2	. . . . . . 7 |- ( ( ph /\ x = A ) -> B e. V )
18		eqid 2083	. . . . . . . 8 |- ( x e. D |-> B ) = ( x e. D |-> B )
19	18	fvmpt2 5307	. . . . . . 7 |- ( ( x e. D /\ B e. V ) -> ( ( x e. D |-> B ) ` x ) = B )
20	16, 17, 19	syl2anc 403	. . . . . 6 |- ( ( ph /\ x = A ) -> ( ( x e. D |-> B ) ` x ) = B )
21	14, 20	eqtr3d 2117	. . . . 5 |- ( ( ph /\ x = A ) -> ( ( x e. D |-> B ) ` A ) = B )
22	21	eqeq2d 2094	. . . 4 |- ( ( ph /\ x = A ) -> ( ( F ` A ) = ( ( x e. D |-> B ) ` A ) <-> ( F ` A ) = B ) )
23		fvmptdf.3	. . . 4 |- ( ( ph /\ x = A ) -> ( ( F ` A ) = B -> ps ) )
24	22, 23	sylbid 148	. . 3 |- ( ( ph /\ x = A ) -> ( ( F ` A ) = ( ( x e. D |-> B ) ` A ) -> ps ) )
25	12, 24	syl5 32	. 2 |- ( ( ph /\ x = A ) -> ( F = ( x e. D |-> B ) -> ps ) )
26	1, 6, 11, 25	exlimdd 1795	1 |- ( ph -> ( F = ( x e. D |-> B ) -> ps ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1285   F/wnf 1390   E.wex 1422   e. wcel 1434   F/_wnfc 2210   _Vcvv 2610   |-> cmpt 3860   ` cfv 4953

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3917  ax-pow 3969  ax-pr 3993

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-sbc 2826  df-csb 2919  df-un 2987  df-in 2989  df-ss 2996  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-br 3807  df-opab 3861  df-mpt 3862  df-id 4077  df-xp 4398  df-rel 4399  df-cnv 4400  df-co 4401  df-dm 4402  df-iota 4918  df-fun 4955  df-fv 4961

This theorem is referenced by:  fvmptdv  5312