Theorem fmpt3d 5714

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 5713	. 2 |- ( ph -> ( x e. A |-> B ) : A --> C )
3		fmpt3d.1	. . 3 |- ( ph -> F = ( x e. A |-> B ) )
4	3	feq1d 5390	. 2 |- ( ph -> ( F : A --> C <-> ( x e. A |-> B ) : A --> C ) )
5	2, 4	mpbird 167	1 |- ( ph -> F : A --> C )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   |-> cmpt 4090   -->wf 5250

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-fv 5262

This theorem is referenced by:  1arithlem3  12503  grpinvf  13119  lspf  13885  bj-charfun  15299  bj-charfundc  15300