Theorem fmpt3d 5569

Description: Domain and codomain of the mapping operation; deduction form. (Contributed by Thierry Arnoux, 4-Jun-2017.)

Hypotheses

Ref	Expression
fmpt3d.1	|- ( ph -> F = ( x e. A |-> B ) )
fmpt3d.2	|- ( ( ph /\ x e. A ) -> B e. C )

Assertion

Ref	Expression
fmpt3d	|- ( ph -> F : A --> C )

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

Allowed substitution hints:   B( x)   F( x)

Proof of Theorem fmpt3d

Step	Hyp	Ref	Expression
1		fmpt3d.2	. . 3 |- ( ( ph /\ x e. A ) -> B e. C )
2	1	fmpttd 5568	. 2 |- ( ph -> ( x e. A |-> B ) : A --> C )
3		fmpt3d.1	. . 3 |- ( ph -> F = ( x e. A |-> B ) )
4	3	feq1d 5254	. 2 |- ( ph -> ( F : A --> C <-> ( x e. A |-> B ) : A --> C ) )
5	2, 4	mpbird 166	1 |- ( ph -> F : A --> C )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   |-> cmpt 3984   -->wf 5114

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-fv 5126

This theorem is referenced by: (None)