Theorem fvmptdv2 5668

Description: Alternate deduction version of fvmpt 5655, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)

Hypotheses

Ref	Expression
fvmptdv2.1	|- ( ph -> A e. D )
fvmptdv2.2	|- ( ( ph /\ x = A ) -> B e. V )
fvmptdv2.3	|- ( ( ph /\ x = A ) -> B = C )

Assertion

Ref	Expression
fvmptdv2	|- ( ph -> ( F = ( x e. D |-> B ) -> ( F ` A ) = C ) )

Distinct variable groups:   x, A   x, C   x, D   ph, x

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

Proof of Theorem fvmptdv2

Step	Hyp	Ref	Expression
1		eqidd 2205	. . 3 |- ( ph -> ( x e. D |-> B ) = ( x e. D |-> B ) )
2		fvmptdv2.3	. . 3 |- ( ( ph /\ x = A ) -> B = C )
3		fvmptdv2.1	. . 3 |- ( ph -> A e. D )
4		elex 2782	. . . . . 6 |- ( A e. D -> A e. _V )
5	3, 4	syl 14	. . . . 5 |- ( ph -> A e. _V )
6		isset 2777	. . . . 5 |- ( A e. _V <-> E. x x = A )
7	5, 6	sylib 122	. . . 4 |- ( ph -> E. x x = A )
8		fvmptdv2.2	. . . . . 6 |- ( ( ph /\ x = A ) -> B e. V )
9		elex 2782	. . . . . 6 |- ( B e. V -> B e. _V )
10	8, 9	syl 14	. . . . 5 |- ( ( ph /\ x = A ) -> B e. _V )
11	2, 10	eqeltrrd 2282	. . . 4 |- ( ( ph /\ x = A ) -> C e. _V )
12	7, 11	exlimddv 1921	. . 3 |- ( ph -> C e. _V )
13	1, 2, 3, 12	fvmptd 5659	. 2 |- ( ph -> ( ( x e. D |-> B ) ` A ) = C )
14		fveq1 5574	. . 3 |- ( F = ( x e. D |-> B ) -> ( F ` A ) = ( ( x e. D |-> B ) ` A ) )
15	14	eqeq1d 2213	. 2 |- ( F = ( x e. D |-> B ) -> ( ( F ` A ) = C <-> ( ( x e. D |-> B ) ` A ) = C ) )
16	13, 15	syl5ibrcom 157	1 |- ( ph -> ( F = ( x e. D |-> B ) -> ( F ` A ) = C ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1372   E.wex 1514   e. wcel 2175   _Vcvv 2771   |-> cmpt 4104   ` cfv 5270

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-sbc 2998  df-csb 3093  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-iota 5231  df-fun 5272  df-fv 5278

This theorem is referenced by: (None)