Theorem fmpt2d 5327

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 2392	. . . 4 |- ( ph -> A. x e. A B e. V )
3		eqid 2040	. . . . 5 |- ( x e. A |-> B ) = ( x e. A |-> B )
4	3	fnmpt 5025	. . . 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 4989	. . 3 |- ( ph -> ( F Fn A <-> ( x e. A |-> B ) Fn A ) )
8	5, 7	mpbird 156	. 2 |- ( ph -> F Fn A )
9		fmpt2d.3	. . 3 |- ( ( ph /\ y e. A ) -> ( F ` y ) e. C )
10	9	ralrimiva 2392	. 2 |- ( ph -> A. y e. A ( F ` y ) e. C )
11		ffnfv 5323	. 2 |- ( F : A --> C <-> ( F Fn A /\ A. y e. A ( F ` y ) e. C ) )
12	8, 10, 11	sylanbrc 394	1 |- ( ph -> F : A --> C )

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1243   e. wcel 1393   A.wral 2306   |-> cmpt 3818   Fn wfn 4897   -->wf 4898   ` cfv 4902

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-fv 4910

This theorem is referenced by: (None)