Theorem fvmptdv2 5575

Description: Alternate deduction version of fvmpt 5563, 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 2166	. . 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 2737	. . . . . 6 |- ( A e. D -> A e. _V )
5	3, 4	syl 14	. . . . 5 |- ( ph -> A e. _V )
6		isset 2732	. . . . 5 |- ( A e. _V <-> E. x x = A )
7	5, 6	sylib 121	. . . 4 |- ( ph -> E. x x = A )
8		fvmptdv2.2	. . . . . 6 |- ( ( ph /\ x = A ) -> B e. V )
9		elex 2737	. . . . . 6 |- ( B e. V -> B e. _V )
10	8, 9	syl 14	. . . . 5 |- ( ( ph /\ x = A ) -> B e. _V )
11	2, 10	eqeltrrd 2244	. . . 4 |- ( ( ph /\ x = A ) -> C e. _V )
12	7, 11	exlimddv 1886	. . 3 |- ( ph -> C e. _V )
13	1, 2, 3, 12	fvmptd 5567	. 2 |- ( ph -> ( ( x e. D |-> B ) ` A ) = C )
14		fveq1 5485	. . 3 |- ( F = ( x e. D |-> B ) -> ( F ` A ) = ( ( x e. D |-> B ) ` A ) )
15	14	eqeq1d 2174	. 2 |- ( F = ( x e. D |-> B ) -> ( ( F ` A ) = C <-> ( ( x e. D |-> B ) ` A ) = C ) )
16	13, 15	syl5ibrcom 156	1 |- ( ph -> ( F = ( x e. D |-> B ) -> ( F ` A ) = C ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   E.wex 1480   e. wcel 2136   _Vcvv 2726   |-> cmpt 4043   ` cfv 5188

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196

This theorem is referenced by: (None)