Theorem fmpt3d 6386

Description: Domain and co-domain 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		eqid 2622	. . 3 |- ( x e. A |-> B ) = ( x e. A |-> B )
3	1, 2	fmptd 6385	. 2 |- ( ph -> ( x e. A |-> B ) : A --> C )
4		fmpt3d.1	. . 3 |- ( ph -> F = ( x e. A |-> B ) )
5	4	feq1d 6030	. 2 |- ( ph -> ( F : A --> C <-> ( x e. A |-> B ) : A --> C ) )
6	3, 5	mpbird 247	1 |- ( ph -> F : A --> C )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   |-> cmpt 4729   -->wf 5884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896

This theorem is referenced by:  fmptco  6396  nmof  22523  ofoprabco  29464  sgnsf  29729  qqhf  30030  indf  30077  esumcocn  30142  ofcf  30165  mbfmcst  30321  dstrvprob  30533  dstfrvclim1  30539  signstf  30643  fsovfd  38306  dssmapnvod  38314  binomcxplemnotnn0  38555  sge0seq  40663  hoicvrrex  40770