Theorem homacd 16691

Description: The codomain of an arrow with known domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)

Hypothesis

Ref	Expression
homahom.h	|- H = (Homa ` C )

Assertion

Ref	Expression
homacd	|- ( F e. ( X H Y ) -> (coda ` F ) = Y )

Proof of Theorem homacd

Step	Hyp	Ref	Expression
1		df-coda 16675	. . . 4 |- coda = ( 2nd o. 1st )
2	1	fveq1i 6192	. . 3 |- (coda ` F ) = ( ( 2nd o. 1st ) ` F )
3		fo1st 7188	. . . . 5 |- 1st : _V -onto-> _V
4		fof 6115	. . . . 5 |- ( 1st : _V -onto-> _V -> 1st : _V --> _V )
5	3, 4	ax-mp 5	. . . 4 |- 1st : _V --> _V
6		elex 3212	. . . 4 |- ( F e. ( X H Y ) -> F e. _V )
7		fvco3 6275	. . . 4 |- ( ( 1st : _V --> _V /\ F e. _V ) -> ( ( 2nd o. 1st ) ` F ) = ( 2nd ` ( 1st ` F ) ) )
8	5, 6, 7	sylancr 695	. . 3 |- ( F e. ( X H Y ) -> ( ( 2nd o. 1st ) ` F ) = ( 2nd ` ( 1st ` F ) ) )
9	2, 8	syl5eq 2668	. 2 |- ( F e. ( X H Y ) -> (coda ` F ) = ( 2nd ` ( 1st ` F ) ) )
10		homahom.h	. . . . . 6 |- H = (Homa ` C )
11	10	homarel 16686	. . . . 5 |- Rel ( X H Y )
12		1st2ndbr 7217	. . . . 5 |- ( ( Rel ( X H Y ) /\ F e. ( X H Y ) ) -> ( 1st ` F ) ( X H Y ) ( 2nd ` F ) )
13	11, 12	mpan 706	. . . 4 |- ( F e. ( X H Y ) -> ( 1st ` F ) ( X H Y ) ( 2nd ` F ) )
14	10	homa1 16687	. . . 4 |- ( ( 1st ` F ) ( X H Y ) ( 2nd ` F ) -> ( 1st ` F ) = <. X , Y >. )
15	13, 14	syl 17	. . 3 |- ( F e. ( X H Y ) -> ( 1st ` F ) = <. X , Y >. )
16	15	fveq2d 6195	. 2 |- ( F e. ( X H Y ) -> ( 2nd ` ( 1st ` F ) ) = ( 2nd ` <. X , Y >. ) )
17		eqid 2622	. . . 4 |- ( Base ` C ) = ( Base ` C )
18	10, 17	homarcl2 16685	. . 3 |- ( F e. ( X H Y ) -> ( X e. ( Base ` C ) /\ Y e. ( Base ` C ) ) )
19		op2ndg 7181	. . 3 |- ( ( X e. ( Base ` C ) /\ Y e. ( Base ` C ) ) -> ( 2nd ` <. X , Y >. ) = Y )
20	18, 19	syl 17	. 2 |- ( F e. ( X H Y ) -> ( 2nd ` <. X , Y >. ) = Y )
21	9, 16, 20	3eqtrd 2660	1 |- ( F e. ( X H Y ) -> (coda ` F ) = Y )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   <.cop 4183   class class class wbr 4653   o. ccom 5118   Rel wrel 5119   -->wf 5884   -onto->wfo 5886   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   Basecbs 15857  cod_accoda 16671  Hom_achoma 16673

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-1st 7168  df-2nd 7169  df-coda 16675  df-homa 16676

This theorem is referenced by:  arwhoma  16695  idacd  16712  homdmcoa  16717  coaval  16718