Theorem fvmptdf 5765

Description: Alternate deduction version of fvmpt 5754, 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 1577	. 2 |- F/ x ph
2		fvmptdf.4	. . . 4 |- F/_ x F
3		nfmpt1 4203	. . . 4 |- F/_ x ( x e. D |-> B )
4	2, 3	nfeq 2392	. . 3 |- F/ x F = ( x e. D |-> B )
5		fvmptdf.5	. . 3 |- F/ x ps
6	4, 5	nfim 1621	. 2 |- F/ x ( F = ( x e. D |-> B ) -> ps )
7		fvmptdf.1	. . . 4 |- ( ph -> A e. D )
8		elex 2825	. . . 4 |- ( A e. D -> A e. _V )
9	7, 8	syl 14	. . 3 |- ( ph -> A e. _V )
10		isset 2820	. . 3 |- ( A e. _V <-> E. x x = A )
11	9, 10	sylib 122	. 2 |- ( ph -> E. x x = A )
12		fveq1 5669	. . 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 5674	. . . . . 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 2309	. . . . . . 7 |- ( ( ph /\ x = A ) -> x e. D )
17		fvmptdf.2	. . . . . . 7 |- ( ( ph /\ x = A ) -> B e. V )
18		eqid 2232	. . . . . . . 8 |- ( x e. D |-> B ) = ( x e. D |-> B )
19	18	fvmpt2 5761	. . . . . . 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 2267	. . . . 5 |- ( ( ph /\ x = A ) -> ( ( x e. D |-> B ) ` A ) = B )
22	21	eqeq2d 2244	. . . 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 1921	1 |- ( ph -> ( F = ( x e. D |-> B ) -> ps ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   F/wnf 1509   E.wex 1541   e. wcel 2203   F/_wnfc 2371   _Vcvv 2813   |-> cmpt 4171   ` cfv 5352

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-csb 3139  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-iota 5312  df-fun 5354  df-fv 5360

This theorem is referenced by:  fvmptdv  5766