Theorem fmpt2d 5658

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 2543	. . . 4 |- ( ph -> A. x e. A B e. V )
3		eqid 2170	. . . . 5 |- ( x e. A |-> B ) = ( x e. A |-> B )
4	3	fnmpt 5324	. . . 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 5288	. . 3 |- ( ph -> ( F Fn A <-> ( x e. A |-> B ) Fn A ) )
8	5, 7	mpbird 166	. 2 |- ( ph -> F Fn A )
9		fmpt2d.3	. . 3 |- ( ( ph /\ y e. A ) -> ( F ` y ) e. C )
10	9	ralrimiva 2543	. 2 |- ( ph -> A. y e. A ( F ` y ) e. C )
11		ffnfv 5654	. 2 |- ( F : A --> C <-> ( F Fn A /\ A. y e. A ( F ` y ) e. C ) )
12	8, 10, 11	sylanbrc 415	1 |- ( ph -> F : A --> C )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   A.wral 2448   |-> cmpt 4050   Fn wfn 5193   -->wf 5194   ` cfv 5198

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206

This theorem is referenced by: (None)