Theorem fvmptdv2 5428

Description: Alternate deduction version of fvmpt 5416, 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 2096	. . 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 2644	. . . . . 6 |- ( A e. D -> A e. _V )
5	3, 4	syl 14	. . . . 5 |- ( ph -> A e. _V )
6		isset 2639	. . . . 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 2644	. . . . . 6 |- ( B e. V -> B e. _V )
10	8, 9	syl 14	. . . . 5 |- ( ( ph /\ x = A ) -> B e. _V )
11	2, 10	eqeltrrd 2172	. . . 4 |- ( ( ph /\ x = A ) -> C e. _V )
12	7, 11	exlimddv 1833	. . 3 |- ( ph -> C e. _V )
13	1, 2, 3, 12	fvmptd 5420	. 2 |- ( ph -> ( ( x e. D |-> B ) ` A ) = C )
14		fveq1 5339	. . 3 |- ( F = ( x e. D |-> B ) -> ( F ` A ) = ( ( x e. D |-> B ) ` A ) )
15	14	eqeq1d 2103	. 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 1296   E.wex 1433   e. wcel 1445   _Vcvv 2633   |-> cmpt 3921   ` cfv 5049

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060

This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-sbc 2855  df-csb 2948  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-iota 5014  df-fun 5051  df-fv 5057

This theorem is referenced by: (None)