Theorem fvmptdf 5605

Description: Alternate deduction version of fvmpt 5595, 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 1528	. 2 |- F/ x ph
2		fvmptdf.4	. . . 4 |- F/_ x F
3		nfmpt1 4098	. . . 4 |- F/_ x ( x e. D |-> B )
4	2, 3	nfeq 2327	. . 3 |- F/ x F = ( x e. D |-> B )
5		fvmptdf.5	. . 3 |- F/ x ps
6	4, 5	nfim 1572	. 2 |- F/ x ( F = ( x e. D |-> B ) -> ps )
7		fvmptdf.1	. . . 4 |- ( ph -> A e. D )
8		elex 2750	. . . 4 |- ( A e. D -> A e. _V )
9	7, 8	syl 14	. . 3 |- ( ph -> A e. _V )
10		isset 2745	. . 3 |- ( A e. _V <-> E. x x = A )
11	9, 10	sylib 122	. 2 |- ( ph -> E. x x = A )
12		fveq1 5516	. . 3 |- ( F = ( x e. D |-> B ) -> ( F ` A ) = ( ( x e. D |-> B ) ` A ) )
13		simpr 110	. . . . . . 7 |- ( ( ph /\ x = A ) -> x = A )
14	13	fveq2d 5521	. . . . . 6 |- ( ( ph /\ x = A ) -> ( ( x e. D |-> B ) ` x ) = ( ( x e. D |-> B ) ` A ) )
15	7	adantr 276	. . . . . . . 8 |- ( ( ph /\ x = A ) -> A e. D )
16	13, 15	eqeltrd 2254	. . . . . . 7 |- ( ( ph /\ x = A ) -> x e. D )
17		fvmptdf.2	. . . . . . 7 |- ( ( ph /\ x = A ) -> B e. V )
18		eqid 2177	. . . . . . . 8 |- ( x e. D |-> B ) = ( x e. D |-> B )
19	18	fvmpt2 5601	. . . . . . 7 |- ( ( x e. D /\ B e. V ) -> ( ( x e. D |-> B ) ` x ) = B )
20	16, 17, 19	syl2anc 411	. . . . . 6 |- ( ( ph /\ x = A ) -> ( ( x e. D |-> B ) ` x ) = B )
21	14, 20	eqtr3d 2212	. . . . 5 |- ( ( ph /\ x = A ) -> ( ( x e. D |-> B ) ` A ) = B )
22	21	eqeq2d 2189	. . . 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 150	. . 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 1872	1 |- ( ph -> ( F = ( x e. D |-> B ) -> ps ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   F/wnf 1460   E.wex 1492   e. wcel 2148   F/_wnfc 2306   _Vcvv 2739   |-> cmpt 4066   ` cfv 5218

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226

This theorem is referenced by:  fvmptdv  5606