Theorem fmpt2d 5348

Description: Domain and codomain of the mapping operation; deduction form. (Contributed by NM, 27-Dec-2014.)

Hypotheses

Ref	Expression
fmpt2d.2	|- ( ( ph /\ x e. A ) -> B e. V )
fmpt2d.1	|- ( ph -> F = ( x e. A |-> B ) )
fmpt2d.3	|- ( ( ph /\ y e. A ) -> ( F ` y ) e. C )

Assertion

Ref	Expression
fmpt2d	|- ( ph -> F : A --> C )

Distinct variable groups:   x, A   y, A   y, C   y, F   ph, x   ph, y

Allowed substitution hints:   B( x, y)   C( x)   F( x)   V( x, y)

Proof of Theorem fmpt2d

Step	Hyp	Ref	Expression
1		fmpt2d.2	. . . . 5 |- ( ( ph /\ x e. A ) -> B e. V )
2	1	ralrimiva 2434	. . . 4 |- ( ph -> A. x e. A B e. V )
3		eqid 2081	. . . . 5 |- ( x e. A |-> B ) = ( x e. A |-> B )
4	3	fnmpt 5045	. . . 4 |- ( A. x e. A B e. V -> ( x e. A |-> B ) Fn A )
5	2, 4	syl 14	. . 3 |- ( ph -> ( x e. A |-> B ) Fn A )
6		fmpt2d.1	. . . 4 |- ( ph -> F = ( x e. A |-> B ) )
7	6	fneq1d 5009	. . 3 |- ( ph -> ( F Fn A <-> ( x e. A |-> B ) Fn A ) )
8	5, 7	mpbird 165	. 2 |- ( ph -> F Fn A )
9		fmpt2d.3	. . 3 |- ( ( ph /\ y e. A ) -> ( F ` y ) e. C )
10	9	ralrimiva 2434	. 2 |- ( ph -> A. y e. A ( F ` y ) e. C )
11		ffnfv 5344	. 2 |- ( F : A --> C <-> ( F Fn A /\ A. y e. A ( F ` y ) e. C ) )
12	8, 10, 11	sylanbrc 408	1 |- ( ph -> F : A --> C )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433   A.wral 2348   |-> cmpt 3839   Fn wfn 4917   -->wf 4918   ` cfv 4922

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-fv 4930

This theorem is referenced by: (None)