Theorem funcid 16530

Description: A functor maps each identity to the corresponding identity in the target category. (Contributed by Mario Carneiro, 2-Jan-2017.)

Hypotheses

Ref	Expression
funcid.b	|- B = ( Base ` D )
funcid.1	|- .1. = ( Id ` D )
funcid.i	|- I = ( Id ` E )
funcid.f	|- ( ph -> F ( D Func E ) G )
funcid.x	|- ( ph -> X e. B )

Assertion

Ref	Expression
funcid	|- ( ph -> ( ( X G X ) ` ( .1. ` X ) ) = ( I ` ( F ` X ) ) )

Proof of Theorem funcid

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

Step	Hyp	Ref	Expression
1		funcid.x	. 2 |- ( ph -> X e. B )
2		funcid.f	. . . . 5 |- ( ph -> F ( D Func E ) G )
3		funcid.b	. . . . . 6 |- B = ( Base ` D )
4		eqid 2622	. . . . . 6 |- ( Base ` E ) = ( Base ` E )
5		eqid 2622	. . . . . 6 |- ( Hom ` D ) = ( Hom ` D )
6		eqid 2622	. . . . . 6 |- ( Hom ` E ) = ( Hom ` E )
7		funcid.1	. . . . . 6 |- .1. = ( Id ` D )
8		funcid.i	. . . . . 6 |- I = ( Id ` E )
9		eqid 2622	. . . . . 6 |- (comp ` D ) = (comp ` D )
10		eqid 2622	. . . . . 6 |- (comp ` E ) = (comp ` E )
11		df-br 4654	. . . . . . . . 9 |- ( F ( D Func E ) G <-> <. F , G >. e. ( D Func E ) )
12	2, 11	sylib 208	. . . . . . . 8 |- ( ph -> <. F , G >. e. ( D Func E ) )
13		funcrcl 16523	. . . . . . . 8 |- ( <. F , G >. e. ( D Func E ) -> ( D e. Cat /\ E e. Cat ) )
14	12, 13	syl 17	. . . . . . 7 |- ( ph -> ( D e. Cat /\ E e. Cat ) )
15	14	simpld 475	. . . . . 6 |- ( ph -> D e. Cat )
16	14	simprd 479	. . . . . 6 |- ( ph -> E e. Cat )
17	3, 4, 5, 6, 7, 8, 9, 10, 15, 16	isfunc 16524	. . . . 5 |- ( ph -> ( F ( D Func E ) G <-> ( F : B --> ( Base ` E ) /\ G e. X_ z e. ( B X. B ) ( ( ( F ` ( 1st ` z ) ) ( Hom ` E ) ( F ` ( 2nd ` z ) ) ) ^m ( ( Hom ` D ) ` z ) ) /\ A. x e. B ( ( ( x G x ) ` ( .1. ` x ) ) = ( I ` ( F ` x ) ) /\ A. y e. B A. z e. B A. m e. ( x ( Hom ` D ) y ) A. n e. ( y ( Hom ` D ) 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	2, 17	mpbid 222	. . . 4 |- ( ph -> ( F : B --> ( Base ` E ) /\ G e. X_ z e. ( B X. B ) ( ( ( F ` ( 1st ` z ) ) ( Hom ` E ) ( F ` ( 2nd ` z ) ) ) ^m ( ( Hom ` D ) ` z ) ) /\ A. x e. B ( ( ( x G x ) ` ( .1. ` x ) ) = ( I ` ( F ` x ) ) /\ A. y e. B A. z e. B A. m e. ( x ( Hom ` D ) y ) A. n e. ( y ( Hom ` D ) 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 ) ) ) ) )
19	18	simp3d 1075	. . 3 |- ( ph -> A. x e. B ( ( ( x G x ) ` ( .1. ` x ) ) = ( I ` ( F ` x ) ) /\ A. y e. B A. z e. B A. m e. ( x ( Hom ` D ) y ) A. n e. ( y ( Hom ` D ) 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 ) ) ) )
20		simpl 473	. . . 4 |- ( ( ( ( x G x ) ` ( .1. ` x ) ) = ( I ` ( F ` x ) ) /\ A. y e. B A. z e. B A. m e. ( x ( Hom ` D ) y ) A. n e. ( y ( Hom ` D ) 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 ) ) ) -> ( ( x G x ) ` ( .1. ` x ) ) = ( I ` ( F ` x ) ) )
21	20	ralimi 2952	. . 3 |- ( A. x e. B ( ( ( x G x ) ` ( .1. ` x ) ) = ( I ` ( F ` x ) ) /\ A. y e. B A. z e. B A. m e. ( x ( Hom ` D ) y ) A. n e. ( y ( Hom ` D ) 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 ) ) ) -> A. x e. B ( ( x G x ) ` ( .1. ` x ) ) = ( I ` ( F ` x ) ) )
22	19, 21	syl 17	. 2 |- ( ph -> A. x e. B ( ( x G x ) ` ( .1. ` x ) ) = ( I ` ( F ` x ) ) )
23		id 22	. . . . . 6 |- ( x = X -> x = X )
24	23, 23	oveq12d 6668	. . . . 5 |- ( x = X -> ( x G x ) = ( X G X ) )
25		fveq2 6191	. . . . 5 |- ( x = X -> ( .1. ` x ) = ( .1. ` X ) )
26	24, 25	fveq12d 6197	. . . 4 |- ( x = X -> ( ( x G x ) ` ( .1. ` x ) ) = ( ( X G X ) ` ( .1. ` X ) ) )
27		fveq2 6191	. . . . 5 |- ( x = X -> ( F ` x ) = ( F ` X ) )
28	27	fveq2d 6195	. . . 4 |- ( x = X -> ( I ` ( F ` x ) ) = ( I ` ( F ` X ) ) )
29	26, 28	eqeq12d 2637	. . 3 |- ( x = X -> ( ( ( x G x ) ` ( .1. ` x ) ) = ( I ` ( F ` x ) ) <-> ( ( X G X ) ` ( .1. ` X ) ) = ( I ` ( F ` X ) ) ) )
30	29	rspcv 3305	. 2 |- ( X e. B -> ( A. x e. B ( ( x G x ) ` ( .1. ` x ) ) = ( I ` ( F ` x ) ) -> ( ( X G X ) ` ( .1. ` X ) ) = ( I ` ( F ` X ) ) ) )
31	1, 22, 30	sylc 65	1 |- ( ph -> ( ( X G X ) ` ( .1. ` X ) ) = ( I ` ( F ` X ) ) )

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:  funcsect  16532  funcoppc  16535  cofucl  16548  funcres  16556  fthsect  16585  catcisolem  16756  prfcl  16843  evlfcl  16862  curf1cl  16868  curfcl  16872  curfuncf  16878  yonedainv  16921