Theorem fvmptdf 5279

Description: Alternate deduction version of fvmpt 5270, 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 1461	. 2 |- F/ x ph
2		fvmptdf.4	. . . 4 |- F/_ x F
3		nfmpt1 3871	. . . 4 |- F/_ x ( x e. D |-> B )
4	2, 3	nfeq 2226	. . 3 |- F/ x F = ( x e. D |-> B )
5		fvmptdf.5	. . 3 |- F/ x ps
6	4, 5	nfim 1504	. 2 |- F/ x ( F = ( x e. D |-> B ) -> ps )
7		fvmptdf.1	. . . 4 |- ( ph -> A e. D )
8		elex 2610	. . . 4 |- ( A e. D -> A e. _V )
9	7, 8	syl 14	. . 3 |- ( ph -> A e. _V )
10		isset 2605	. . 3 |- ( A e. _V <-> E. x x = A )
11	9, 10	sylib 120	. 2 |- ( ph -> E. x x = A )
12		fveq1 5197	. . 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 5202	. . . . . 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 2155	. . . . . . 7 |- ( ( ph /\ x = A ) -> x e. D )
17		fvmptdf.2	. . . . . . 7 |- ( ( ph /\ x = A ) -> B e. V )
18		eqid 2081	. . . . . . . 8 |- ( x e. D |-> B ) = ( x e. D |-> B )
19	18	fvmpt2 5275	. . . . . . 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 2115	. . . . 5 |- ( ( ph /\ x = A ) -> ( ( x e. D |-> B ) ` A ) = B )
22	21	eqeq2d 2092	. . . 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 1793	1 |- ( ph -> ( F = ( x e. D |-> B ) -> ps ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   F/wnf 1389   E.wex 1421   e. wcel 1433   F/_wnfc 2206   _Vcvv 2601   |-> cmpt 3839   ` cfv 4922

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-csb 2909  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-iota 4887  df-fun 4924  df-fv 4930

This theorem is referenced by:  fvmptdv  5280