Theorem fmpt2d 5839

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 2615	. . . 4 |- ( ph -> A. x e. A B e. V )
3		eqid 2232	. . . . 5 |- ( x e. A |-> B ) = ( x e. A |-> B )
4	3	fnmpt 5485	. . . 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 5446	. . 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 2615	. 2 |- ( ph -> A. y e. A ( F ` y ) e. C )
11		ffnfv 5835	. 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 2203   A.wral 2520   |-> cmpt 4171   Fn wfn 5347   -->wf 5348   ` cfv 5352

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 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-fv 5360

This theorem is referenced by: (None)