Theorem fvmptdf 5516

Description: Alternate deduction version of fvmpt 5506, 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 1509	. 2 |- F/ x ph
2		fvmptdf.4	. . . 4 |- F/_ x F
3		nfmpt1 4029	. . . 4 |- F/_ x ( x e. D |-> B )
4	2, 3	nfeq 2290	. . 3 |- F/ x F = ( x e. D |-> B )
5		fvmptdf.5	. . 3 |- F/ x ps
6	4, 5	nfim 1552	. 2 |- F/ x ( F = ( x e. D |-> B ) -> ps )
7		fvmptdf.1	. . . 4 |- ( ph -> A e. D )
8		elex 2700	. . . 4 |- ( A e. D -> A e. _V )
9	7, 8	syl 14	. . 3 |- ( ph -> A e. _V )
10		isset 2695	. . 3 |- ( A e. _V <-> E. x x = A )
11	9, 10	sylib 121	. 2 |- ( ph -> E. x x = A )
12		fveq1 5428	. . 3 |- ( F = ( x e. D |-> B ) -> ( F ` A ) = ( ( x e. D |-> B ) ` A ) )
13		simpr 109	. . . . . . 7 |- ( ( ph /\ x = A ) -> x = A )
14	13	fveq2d 5433	. . . . . 6 |- ( ( ph /\ x = A ) -> ( ( x e. D |-> B ) ` x ) = ( ( x e. D |-> B ) ` A ) )
15	7	adantr 274	. . . . . . . 8 |- ( ( ph /\ x = A ) -> A e. D )
16	13, 15	eqeltrd 2217	. . . . . . 7 |- ( ( ph /\ x = A ) -> x e. D )
17		fvmptdf.2	. . . . . . 7 |- ( ( ph /\ x = A ) -> B e. V )
18		eqid 2140	. . . . . . . 8 |- ( x e. D |-> B ) = ( x e. D |-> B )
19	18	fvmpt2 5512	. . . . . . 7 |- ( ( x e. D /\ B e. V ) -> ( ( x e. D |-> B ) ` x ) = B )
20	16, 17, 19	syl2anc 409	. . . . . 6 |- ( ( ph /\ x = A ) -> ( ( x e. D |-> B ) ` x ) = B )
21	14, 20	eqtr3d 2175	. . . . 5 |- ( ( ph /\ x = A ) -> ( ( x e. D |-> B ) ` A ) = B )
22	21	eqeq2d 2152	. . . 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 149	. . 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 1845	1 |- ( ph -> ( F = ( x e. D |-> B ) -> ps ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1332   F/wnf 1437   E.wex 1469   e. wcel 1481   F/_wnfc 2269   _Vcvv 2689   |-> cmpt 3997   ` cfv 5131

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-csb 3008  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-iota 5096  df-fun 5133  df-fv 5139

This theorem is referenced by:  fvmptdv  5517