Theorem fvmptd3f 6295

Description: Alternate deduction version of fvmpt 6282 with three non-freeness hypotheses instead of distinct variable conditions. (Contributed by AV, 19-Jan-2022.)

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 ) )
fvmptd3f.4	|- F/_ x F
fvmptd3f.5	|- F/ x ps
fvmptd3f.6	|- F/ x ph

Assertion

Ref	Expression
fvmptd3f	|- ( ph -> ( F = ( x e. D |-> B ) -> ps ) )

Distinct variable groups:   x, A   x, D

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

Proof of Theorem fvmptd3f

Step	Hyp	Ref	Expression
1		fvmptd3f.6	. 2 |- F/ x ph
2		fvmptd3f.4	. . . 4 |- F/_ x F
3		nfmpt1 4747	. . . 4 |- F/_ x ( x e. D |-> B )
4	2, 3	nfeq 2776	. . 3 |- F/ x F = ( x e. D |-> B )
5		fvmptd3f.5	. . 3 |- F/ x ps
6	4, 5	nfim 1825	. 2 |- F/ x ( F = ( x e. D |-> B ) -> ps )
7		fvmptdf.1	. . . 4 |- ( ph -> A e. D )
8	7	elexd 3214	. . 3 |- ( ph -> A e. _V )
9		isset 3207	. . 3 |- ( A e. _V <-> E. x x = A )
10	8, 9	sylib 208	. 2 |- ( ph -> E. x x = A )
11		fveq1 6190	. . 3 |- ( F = ( x e. D |-> B ) -> ( F ` A ) = ( ( x e. D |-> B ) ` A ) )
12		simpr 477	. . . . . . 7 |- ( ( ph /\ x = A ) -> x = A )
13	12	fveq2d 6195	. . . . . 6 |- ( ( ph /\ x = A ) -> ( ( x e. D |-> B ) ` x ) = ( ( x e. D |-> B ) ` A ) )
14	7	adantr 481	. . . . . . . 8 |- ( ( ph /\ x = A ) -> A e. D )
15	12, 14	eqeltrd 2701	. . . . . . 7 |- ( ( ph /\ x = A ) -> x e. D )
16		fvmptdf.2	. . . . . . 7 |- ( ( ph /\ x = A ) -> B e. V )
17		eqid 2622	. . . . . . . 8 |- ( x e. D |-> B ) = ( x e. D |-> B )
18	17	fvmpt2 6291	. . . . . . 7 |- ( ( x e. D /\ B e. V ) -> ( ( x e. D |-> B ) ` x ) = B )
19	15, 16, 18	syl2anc 693	. . . . . 6 |- ( ( ph /\ x = A ) -> ( ( x e. D |-> B ) ` x ) = B )
20	13, 19	eqtr3d 2658	. . . . 5 |- ( ( ph /\ x = A ) -> ( ( x e. D |-> B ) ` A ) = B )
21	20	eqeq2d 2632	. . . 4 |- ( ( ph /\ x = A ) -> ( ( F ` A ) = ( ( x e. D |-> B ) ` A ) <-> ( F ` A ) = B ) )
22		fvmptdf.3	. . . 4 |- ( ( ph /\ x = A ) -> ( ( F ` A ) = B -> ps ) )
23	21, 22	sylbid 230	. . 3 |- ( ( ph /\ x = A ) -> ( ( F ` A ) = ( ( x e. D |-> B ) ` A ) -> ps ) )
24	11, 23	syl5 34	. 2 |- ( ( ph /\ x = A ) -> ( F = ( x e. D |-> B ) -> ps ) )
25	1, 6, 10, 24	exlimdd 2088	1 |- ( ph -> ( F = ( x e. D |-> B ) -> ps ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   F/wnf 1708   e. wcel 1990   F/_wnfc 2751   _Vcvv 3200   |-> cmpt 4729   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896

This theorem is referenced by:  fvmptdf  6296