Theorem homadmcd 16692

Description: Decompose an arrow into domain, codomain, and morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)

Hypothesis

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

Assertion

Ref	Expression
homadmcd	|- ( F e. ( X H Y ) -> F = <. X , Y , ( 2nd ` F ) >. )

Proof of Theorem homadmcd

Step	Hyp	Ref	Expression
1		homahom.h	. . . . 5 |- H = (Homa ` C )
2	1	homarel 16686	. . . 4 |- Rel ( X H Y )
3		1st2nd 7214	. . . 4 |- ( ( Rel ( X H Y ) /\ F e. ( X H Y ) ) -> F = <. ( 1st ` F ) , ( 2nd ` F ) >. )
4	2, 3	mpan 706	. . 3 |- ( F e. ( X H Y ) -> F = <. ( 1st ` F ) , ( 2nd ` F ) >. )
5		1st2ndbr 7217	. . . . . 6 |- ( ( Rel ( X H Y ) /\ F e. ( X H Y ) ) -> ( 1st ` F ) ( X H Y ) ( 2nd ` F ) )
6	2, 5	mpan 706	. . . . 5 |- ( F e. ( X H Y ) -> ( 1st ` F ) ( X H Y ) ( 2nd ` F ) )
7	1	homa1 16687	. . . . 5 |- ( ( 1st ` F ) ( X H Y ) ( 2nd ` F ) -> ( 1st ` F ) = <. X , Y >. )
8	6, 7	syl 17	. . . 4 |- ( F e. ( X H Y ) -> ( 1st ` F ) = <. X , Y >. )
9	8	opeq1d 4408	. . 3 |- ( F e. ( X H Y ) -> <. ( 1st ` F ) , ( 2nd ` F ) >. = <. <. X , Y >. , ( 2nd ` F ) >. )
10	4, 9	eqtrd 2656	. 2 |- ( F e. ( X H Y ) -> F = <. <. X , Y >. , ( 2nd ` F ) >. )
11		df-ot 4186	. 2 |- <. X , Y , ( 2nd ` F ) >. = <. <. X , Y >. , ( 2nd ` F ) >.
12	10, 11	syl6eqr 2674	1 |- ( F e. ( X H Y ) -> F = <. X , Y , ( 2nd ` F ) >. )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   <.cop 4183   <.cotp 4185   class class class wbr 4653   Rel wrel 5119   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167  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-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-ov 6653  df-1st 7168  df-2nd 7169  df-homa 16676

This theorem is referenced by:  arwdmcd  16702  arwlid  16722  arwrid  16723