Theorem funcixp 16527

Description: The morphism part of a functor is a function on homsets. (Contributed by Mario Carneiro, 2-Jan-2017.)

Hypotheses

Ref	Expression
funcixp.b	|- B = ( Base ` D )
funcixp.h	|- H = ( Hom ` D )
funcixp.j	|- J = ( Hom ` E )
funcixp.f	|- ( ph -> F ( D Func E ) G )

Assertion

Ref	Expression
funcixp	|- ( ph -> G e. X_ z e. ( B X. B ) ( ( ( F ` ( 1st ` z ) ) J ( F ` ( 2nd ` z ) ) ) ^m ( H ` z ) ) )

Distinct variable groups:   z, B   z, D   z, E   ph, z   z, F   z, G   z, J   z, H

Proof of Theorem funcixp

Dummy variables m n x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		funcixp.f	. . 3 |- ( ph -> F ( D Func E ) G )
2		funcixp.b	. . . 4 |- B = ( Base ` D )
3		eqid 2622	. . . 4 |- ( Base ` E ) = ( Base ` E )
4		funcixp.h	. . . 4 |- H = ( Hom ` D )
5		funcixp.j	. . . 4 |- J = ( Hom ` E )
6		eqid 2622	. . . 4 |- ( Id ` D ) = ( Id ` D )
7		eqid 2622	. . . 4 |- ( Id ` E ) = ( Id ` E )
8		eqid 2622	. . . 4 |- (comp ` D ) = (comp ` D )
9		eqid 2622	. . . 4 |- (comp ` E ) = (comp ` E )
10		df-br 4654	. . . . . . 7 |- ( F ( D Func E ) G <-> <. F , G >. e. ( D Func E ) )
11	1, 10	sylib 208	. . . . . 6 |- ( ph -> <. F , G >. e. ( D Func E ) )
12		funcrcl 16523	. . . . . 6 |- ( <. F , G >. e. ( D Func E ) -> ( D e. Cat /\ E e. Cat ) )
13	11, 12	syl 17	. . . . 5 |- ( ph -> ( D e. Cat /\ E e. Cat ) )
14	13	simpld 475	. . . 4 |- ( ph -> D e. Cat )
15	13	simprd 479	. . . 4 |- ( ph -> E e. Cat )
16	2, 3, 4, 5, 6, 7, 8, 9, 14, 15	isfunc 16524	. . 3 |- ( ph -> ( F ( D Func E ) G <-> ( F : B --> ( Base ` E ) /\ G e. X_ z e. ( B X. B ) ( ( ( F ` ( 1st ` z ) ) J ( F ` ( 2nd ` z ) ) ) ^m ( H ` z ) ) /\ A. x e. B ( ( ( x G x ) ` ( ( Id ` D ) ` x ) ) = ( ( Id ` E ) ` ( F ` x ) ) /\ A. y e. B A. z e. B A. m e. ( x H y ) A. n e. ( y H z ) ( ( x G z ) ` ( n ( <. x , y >. (comp ` D ) z ) m ) ) = ( ( ( y G z ) ` n ) ( <. ( F ` x ) , ( F ` y ) >. (comp ` E ) ( F ` z ) ) ( ( x G y ) ` m ) ) ) ) ) )
17	1, 16	mpbid 222	. 2 |- ( ph -> ( F : B --> ( Base ` E ) /\ G e. X_ z e. ( B X. B ) ( ( ( F ` ( 1st ` z ) ) J ( F ` ( 2nd ` z ) ) ) ^m ( H ` z ) ) /\ A. x e. B ( ( ( x G x ) ` ( ( Id ` D ) ` x ) ) = ( ( Id ` E ) ` ( F ` x ) ) /\ A. y e. B A. z e. B A. m e. ( x H y ) A. n e. ( y H z ) ( ( x G z ) ` ( n ( <. x , y >. (comp ` D ) z ) m ) ) = ( ( ( y G z ) ` n ) ( <. ( F ` x ) , ( F ` y ) >. (comp ` E ) ( F ` z ) ) ( ( x G y ) ` m ) ) ) ) )
18	17	simp2d 1074	1 |- ( ph -> G e. X_ z e. ( B X. B ) ( ( ( F ` ( 1st ` z ) ) J ( F ` ( 2nd ` z ) ) ) ^m ( H ` z ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   <.cop 4183   class class class wbr 4653   X. cxp 5112   -->wf 5884   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   ^m cmap 7857   X_cixp 7908   Basecbs 15857   Hom chom 15952  compcco 15953   Catccat 16325   Idccid 16326   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:  funcf2  16528  funcfn2  16529  wunfunc  16559