Theorem fvmptdv2 5795

Description: Alternate deduction version of fvmpt 5782, 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 2239	. . 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 2833	. . . . . 6 |- ( A e. D -> A e. _V )
5	3, 4	syl 14	. . . . 5 |- ( ph -> A e. _V )
6		isset 2828	. . . . 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 2833	. . . . . 6 |- ( B e. V -> B e. _V )
10	8, 9	syl 14	. . . . 5 |- ( ( ph /\ x = A ) -> B e. _V )
11	2, 10	eqeltrrd 2316	. . . 4 |- ( ( ph /\ x = A ) -> C e. _V )
12	7, 11	exlimddv 1954	. . 3 |- ( ph -> C e. _V )
13	1, 2, 3, 12	fvmptd 5786	. 2 |- ( ph -> ( ( x e. D |-> B ) ` A ) = C )
14		fveq1 5694	. . 3 |- ( F = ( x e. D |-> B ) -> ( F ` A ) = ( ( x e. D |-> B ) ` A ) )
15	14	eqeq1d 2247	. 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 ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   E.wex 1545   e. wcel 2209   _Vcvv 2821   |-> cmpt 4192   ` cfv 5377

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-iota 5337  df-fun 5379  df-fv 5385

This theorem is used by: (None)