Theorem ida2 16709

Description: Morphism part of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)

Hypotheses

Ref	Expression
idafval.i	|- I = (Ida ` C )
idafval.b	|- B = ( Base ` C )
idafval.c	|- ( ph -> C e. Cat )
idafval.1	|- .1. = ( Id ` C )
idaval.x	|- ( ph -> X e. B )

Assertion

Ref	Expression
ida2	|- ( ph -> ( 2nd ` ( I ` X ) ) = ( .1. ` X ) )

Proof of Theorem ida2

Step	Hyp	Ref	Expression
1		idafval.i	. . . 4 |- I = (Ida ` C )
2		idafval.b	. . . 4 |- B = ( Base ` C )
3		idafval.c	. . . 4 |- ( ph -> C e. Cat )
4		idafval.1	. . . 4 |- .1. = ( Id ` C )
5		idaval.x	. . . 4 |- ( ph -> X e. B )
6	1, 2, 3, 4, 5	idaval 16708	. . 3 |- ( ph -> ( I ` X ) = <. X , X , ( .1. ` X ) >. )
7	6	fveq2d 6195	. 2 |- ( ph -> ( 2nd ` ( I ` X ) ) = ( 2nd ` <. X , X , ( .1. ` X ) >. ) )
8		fvex 6201	. . 3 |- ( .1. ` X ) e. _V
9		ot3rdg 7184	. . 3 |- ( ( .1. ` X ) e. _V -> ( 2nd ` <. X , X , ( .1. ` X ) >. ) = ( .1. ` X ) )
10	8, 9	ax-mp 5	. 2 |- ( 2nd ` <. X , X , ( .1. ` X ) >. ) = ( .1. ` X )
11	7, 10	syl6eq 2672	1 |- ( ph -> ( 2nd ` ( I ` X ) ) = ( .1. ` X ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   <.cotp 4185   ` cfv 5888   2ndc2nd 7167   Basecbs 15857   Catccat 16325   Idccid 16326  Id_acida 16703

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-sn 4178  df-pr 4180  df-op 4184  df-ot 4186  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-2nd 7169  df-ida 16705

This theorem is referenced by:  arwlid  16722  arwrid  16723