Theorem fvmptdv2 5767

Description: Alternate deduction version of fvmpt 5754, 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 2233	. . 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 2825	. . . . . 6 |- ( A e. D -> A e. _V )
5	3, 4	syl 14	. . . . 5 |- ( ph -> A e. _V )
6		isset 2820	. . . . 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 2825	. . . . . 6 |- ( B e. V -> B e. _V )
10	8, 9	syl 14	. . . . 5 |- ( ( ph /\ x = A ) -> B e. _V )
11	2, 10	eqeltrrd 2310	. . . 4 |- ( ( ph /\ x = A ) -> C e. _V )
12	7, 11	exlimddv 1948	. . 3 |- ( ph -> C e. _V )
13	1, 2, 3, 12	fvmptd 5758	. 2 |- ( ph -> ( ( x e. D |-> B ) ` A ) = C )
14		fveq1 5669	. . 3 |- ( F = ( x e. D |-> B ) -> ( F ` A ) = ( ( x e. D |-> B ) ` A ) )
15	14	eqeq1d 2241	. 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 1398   E.wex 1541   e. wcel 2203   _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: (None)