Theorem idfu2nd 16537

Description: Value of the morphism part of the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)

Hypotheses

Ref	Expression
idfuval.i	|- I = (idfunc ` C )
idfuval.b	|- B = ( Base ` C )
idfuval.c	|- ( ph -> C e. Cat )
idfuval.h	|- H = ( Hom ` C )
idfu2nd.x	|- ( ph -> X e. B )
idfu2nd.y	|- ( ph -> Y e. B )

Assertion

Ref	Expression
idfu2nd	|- ( ph -> ( X ( 2nd ` I ) Y ) = ( _I |` ( X H Y ) ) )

Proof of Theorem idfu2nd

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ov 6653	. 2 |- ( X ( 2nd ` I ) Y ) = ( ( 2nd ` I ) ` <. X , Y >. )
2		idfuval.i	. . . . . 6 |- I = (idfunc ` C )
3		idfuval.b	. . . . . 6 |- B = ( Base ` C )
4		idfuval.c	. . . . . 6 |- ( ph -> C e. Cat )
5		idfuval.h	. . . . . 6 |- H = ( Hom ` C )
6	2, 3, 4, 5	idfuval 16536	. . . . 5 |- ( ph -> I = <. ( _I |` B ) , ( z e. ( B X. B ) |-> ( _I |` ( H ` z ) ) ) >. )
7	6	fveq2d 6195	. . . 4 |- ( ph -> ( 2nd ` I ) = ( 2nd ` <. ( _I |` B ) , ( z e. ( B X. B ) |-> ( _I |` ( H ` z ) ) ) >. ) )
8		fvex 6201	. . . . . . 7 |- ( Base ` C ) e. _V
9	3, 8	eqeltri 2697	. . . . . 6 |- B e. _V
10		resiexg 7102	. . . . . 6 |- ( B e. _V -> ( _I |` B ) e. _V )
11	9, 10	ax-mp 5	. . . . 5 |- ( _I |` B ) e. _V
12	9, 9	xpex 6962	. . . . . 6 |- ( B X. B ) e. _V
13	12	mptex 6486	. . . . 5 |- ( z e. ( B X. B ) |-> ( _I |` ( H ` z ) ) ) e. _V
14	11, 13	op2nd 7177	. . . 4 |- ( 2nd ` <. ( _I |` B ) , ( z e. ( B X. B ) |-> ( _I |` ( H ` z ) ) ) >. ) = ( z e. ( B X. B ) |-> ( _I |` ( H ` z ) ) )
15	7, 14	syl6eq 2672	. . 3 |- ( ph -> ( 2nd ` I ) = ( z e. ( B X. B ) |-> ( _I |` ( H ` z ) ) ) )
16		simpr 477	. . . . . 6 |- ( ( ph /\ z = <. X , Y >. ) -> z = <. X , Y >. )
17	16	fveq2d 6195	. . . . 5 |- ( ( ph /\ z = <. X , Y >. ) -> ( H ` z ) = ( H ` <. X , Y >. ) )
18		df-ov 6653	. . . . 5 |- ( X H Y ) = ( H ` <. X , Y >. )
19	17, 18	syl6eqr 2674	. . . 4 |- ( ( ph /\ z = <. X , Y >. ) -> ( H ` z ) = ( X H Y ) )
20	19	reseq2d 5396	. . 3 |- ( ( ph /\ z = <. X , Y >. ) -> ( _I |` ( H ` z ) ) = ( _I |` ( X H Y ) ) )
21		idfu2nd.x	. . . 4 |- ( ph -> X e. B )
22		idfu2nd.y	. . . 4 |- ( ph -> Y e. B )
23		opelxpi 5148	. . . 4 |- ( ( X e. B /\ Y e. B ) -> <. X , Y >. e. ( B X. B ) )
24	21, 22, 23	syl2anc 693	. . 3 |- ( ph -> <. X , Y >. e. ( B X. B ) )
25		ovex 6678	. . . 4 |- ( X H Y ) e. _V
26		resiexg 7102	. . . 4 |- ( ( X H Y ) e. _V -> ( _I |` ( X H Y ) ) e. _V )
27	25, 26	mp1i 13	. . 3 |- ( ph -> ( _I |` ( X H Y ) ) e. _V )
28	15, 20, 24, 27	fvmptd 6288	. 2 |- ( ph -> ( ( 2nd ` I ) ` <. X , Y >. ) = ( _I |` ( X H Y ) ) )
29	1, 28	syl5eq 2668	1 |- ( ph -> ( X ( 2nd ` I ) Y ) = ( _I |` ( X H Y ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   <.cop 4183   |-> cmpt 4729   _I cid 5023   X. cxp 5112   |` cres 5116   ` cfv 5888  (class class class)co 6650   2ndc2nd 7167   Basecbs 15857   Hom chom 15952   Catccat 16325  id_funccidfu 16515

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-2nd 7169  df-idfu 16519

This theorem is referenced by:  idfu2  16538  idfucl  16541  cofulid  16550  cofurid  16551  idffth  16593  ressffth  16598  catciso  16757