Theorem arwdmcd 16702

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

Hypothesis

Ref	Expression
arwrcl.a	|- A = (Nat ` C )

Assertion

Ref	Expression
arwdmcd	|- ( F e. A -> F = <. (domA ` F ) , (coda ` F ) , ( 2nd ` F ) >. )

Proof of Theorem arwdmcd

Step	Hyp	Ref	Expression
1		arwrcl.a	. . 3 |- A = (Nat ` C )
2		eqid 2622	. . 3 |- (Homa ` C ) = (Homa ` C )
3	1, 2	arwhoma 16695	. 2 |- ( F e. A -> F e. ( (domA ` F ) (Homa ` C ) (coda ` F ) ) )
4	2	homadmcd 16692	. 2 |- ( F e. ( (domA ` F ) (Homa ` C ) (coda ` F ) ) -> F = <. (domA ` F ) , (coda ` F ) , ( 2nd ` F ) >. )
5	3, 4	syl 17	1 |- ( F e. A -> F = <. (domA ` F ) , (coda ` F ) , ( 2nd ` F ) >. )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   <.cotp 4185   ` cfv 5888  (class class class)co 6650   2ndc2nd 7167  domAcdoma 16670  cod_accoda 16671  Natcarw 16672  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-doma 16674  df-coda 16675  df-homa 16676  df-arw 16677

This theorem is referenced by: (None)