Theorem arwhoma 16695

Description: An arrow is contained in the hom-set corresponding to its domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)

Hypotheses

Ref	Expression
arwrcl.a	|- A = (Nat ` C )
arwhoma.h	|- H = (Homa ` C )

Assertion

Ref	Expression
arwhoma	|- ( F e. A -> F e. ( (domA ` F ) H (coda ` F ) ) )

Proof of Theorem arwhoma

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		arwrcl.a	. . . . . . 7 |- A = (Nat ` C )
2		arwhoma.h	. . . . . . 7 |- H = (Homa ` C )
3	1, 2	arwval 16693	. . . . . 6 |- A = U. ran H
4	3	eleq2i 2693	. . . . 5 |- ( F e. A <-> F e. U. ran H )
5	4	biimpi 206	. . . 4 |- ( F e. A -> F e. U. ran H )
6		eqid 2622	. . . . . 6 |- ( Base ` C ) = ( Base ` C )
7	1	arwrcl 16694	. . . . . 6 |- ( F e. A -> C e. Cat )
8	2, 6, 7	homaf 16680	. . . . 5 |- ( F e. A -> H : ( ( Base ` C ) X. ( Base ` C ) ) --> ~P ( ( ( Base ` C ) X. ( Base ` C ) ) X. _V ) )
9		ffn 6045	. . . . 5 |- ( H : ( ( Base ` C ) X. ( Base ` C ) ) --> ~P ( ( ( Base ` C ) X. ( Base ` C ) ) X. _V ) -> H Fn ( ( Base ` C ) X. ( Base ` C ) ) )
10		fnunirn 6511	. . . . 5 |- ( H Fn ( ( Base ` C ) X. ( Base ` C ) ) -> ( F e. U. ran H <-> E. z e. ( ( Base ` C ) X. ( Base ` C ) ) F e. ( H ` z ) ) )
11	8, 9, 10	3syl 18	. . . 4 |- ( F e. A -> ( F e. U. ran H <-> E. z e. ( ( Base ` C ) X. ( Base ` C ) ) F e. ( H ` z ) ) )
12	5, 11	mpbid 222	. . 3 |- ( F e. A -> E. z e. ( ( Base ` C ) X. ( Base ` C ) ) F e. ( H ` z ) )
13		fveq2 6191	. . . . . 6 |- ( z = <. x , y >. -> ( H ` z ) = ( H ` <. x , y >. ) )
14		df-ov 6653	. . . . . 6 |- ( x H y ) = ( H ` <. x , y >. )
15	13, 14	syl6eqr 2674	. . . . 5 |- ( z = <. x , y >. -> ( H ` z ) = ( x H y ) )
16	15	eleq2d 2687	. . . 4 |- ( z = <. x , y >. -> ( F e. ( H ` z ) <-> F e. ( x H y ) ) )
17	16	rexxp 5264	. . 3 |- ( E. z e. ( ( Base ` C ) X. ( Base ` C ) ) F e. ( H ` z ) <-> E. x e. ( Base ` C ) E. y e. ( Base ` C ) F e. ( x H y ) )
18	12, 17	sylib 208	. 2 |- ( F e. A -> E. x e. ( Base ` C ) E. y e. ( Base ` C ) F e. ( x H y ) )
19		id 22	. . . . 5 |- ( F e. ( x H y ) -> F e. ( x H y ) )
20	2	homadm 16690	. . . . . 6 |- ( F e. ( x H y ) -> (domA ` F ) = x )
21	2	homacd 16691	. . . . . 6 |- ( F e. ( x H y ) -> (coda ` F ) = y )
22	20, 21	oveq12d 6668	. . . . 5 |- ( F e. ( x H y ) -> ( (domA ` F ) H (coda ` F ) ) = ( x H y ) )
23	19, 22	eleqtrrd 2704	. . . 4 |- ( F e. ( x H y ) -> F e. ( (domA ` F ) H (coda ` F ) ) )
24	23	rexlimivw 3029	. . 3 |- ( E. y e. ( Base ` C ) F e. ( x H y ) -> F e. ( (domA ` F ) H (coda ` F ) ) )
25	24	rexlimivw 3029	. 2 |- ( E. x e. ( Base ` C ) E. y e. ( Base ` C ) F e. ( x H y ) -> F e. ( (domA ` F ) H (coda ` F ) ) )
26	18, 25	syl 17	1 |- ( F e. A -> F e. ( (domA ` F ) H (coda ` F ) ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   E.wrex 2913   _Vcvv 3200   ~Pcpw 4158   <.cop 4183   U.cuni 4436   X. cxp 5112   ran crn 5115   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   Basecbs 15857  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-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:  arwdm  16697  arwcd  16698  arwhom  16701  arwdmcd  16702  coapm  16721