Theorem fmpt2d 5817

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 2606	. . . 4 |- ( ph -> A. x e. A B e. V )
3		eqid 2231	. . . . 5 |- ( x e. A |-> B ) = ( x e. A |-> B )
4	3	fnmpt 5466	. . . 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 5427	. . 3 |- ( ph -> ( F Fn A <-> ( x e. A |-> B ) Fn A ) )
8	5, 7	mpbird 167	. 2 |- ( ph -> F Fn A )
9		fmpt2d.3	. . 3 |- ( ( ph /\ y e. A ) -> ( F ` y ) e. C )
10	9	ralrimiva 2606	. 2 |- ( ph -> A. y e. A ( F ` y ) e. C )
11		ffnfv 5813	. 2 |- ( F : A --> C <-> ( F Fn A /\ A. y e. A ( F ` y ) e. C ) )
12	8, 10, 11	sylanbrc 417	1 |- ( ph -> F : A --> C )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   A.wral 2511   |-> cmpt 4155   Fn wfn 5328   -->wf 5329   ` cfv 5333

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341

This theorem is referenced by: (None)