Theorem funcfn2 16529

Description: The morphism part of a functor is a function. (Contributed by Mario Carneiro, 3-Jan-2017.)

Hypotheses

Ref	Expression
funcfn2.b	|- B = ( Base ` D )
funcfn2.f	|- ( ph -> F ( D Func E ) G )

Assertion

Ref	Expression
funcfn2	|- ( ph -> G Fn ( B X. B ) )

Proof of Theorem funcfn2

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		funcfn2.b	. . 3 |- B = ( Base ` D )
2		eqid 2622	. . 3 |- ( Hom ` D ) = ( Hom ` D )
3		eqid 2622	. . 3 |- ( Hom ` E ) = ( Hom ` E )
4		funcfn2.f	. . 3 |- ( ph -> F ( D Func E ) G )
5	1, 2, 3, 4	funcixp 16527	. 2 |- ( ph -> G e. X_ x e. ( B X. B ) ( ( ( F ` ( 1st ` x ) ) ( Hom ` E ) ( F ` ( 2nd ` x ) ) ) ^m ( ( Hom ` D ) ` x ) ) )
6		ixpfn 7914	. 2 |- ( G e. X_ x e. ( B X. B ) ( ( ( F ` ( 1st ` x ) ) ( Hom ` E ) ( F ` ( 2nd ` x ) ) ) ^m ( ( Hom ` D ) ` x ) ) -> G Fn ( B X. B ) )
7	5, 6	syl 17	1 |- ( ph -> G Fn ( B X. B ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   class class class wbr 4653   X. cxp 5112   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   ^m cmap 7857   X_cixp 7908   Basecbs 15857   Hom chom 15952   Func cfunc 16514

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

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-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  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-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-ixp 7909  df-func 16518

This theorem is referenced by:  funcoppc  16535  cofuval  16542  cofulid  16550  cofurid  16551  prf1st  16844  prf2nd  16845  1st2ndprf  16846  curfuncf  16878  uncfcurf  16879  curf2ndf  16887