Theorem arwval 15449

Description: The set of arrows is the union of all the disjointified hom-sets. (Contributed by Mario Carneiro, 11-Jan-2017.)

Hypotheses

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

Assertion

Ref	Expression
arwval	|- A = U. ran H

Proof of Theorem arwval

Dummy variables x c are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		arwval.a	. 2 |- A = (Nat ` C )
2		fveq2 5872	. . . . . . 7 |- ( c = C -> (Homa ` c ) = (Homa ` C ) )
3		arwval.h	. . . . . . 7 |- H = (Homa ` C )
4	2, 3	syl6eqr 2516	. . . . . 6 |- ( c = C -> (Homa ` c ) = H )
5	4	rneqd 5240	. . . . 5 |- ( c = C -> ran (Homa ` c ) = ran H )
6	5	unieqd 4261	. . . 4 |- ( c = C -> U. ran (Homa ` c ) = U. ran H )
7		df-arw 15433	. . . 4 |- Nat = ( c e. Cat |-> U. ran (Homa ` c ) )
8		fvex 5882	. . . . . . 7 |- (Homa ` C ) e. _V
9	3, 8	eqeltri 2541	. . . . . 6 |- H e. _V
10	9	rnex 6733	. . . . 5 |- ran H e. _V
11	10	uniex 6595	. . . 4 |- U. ran H e. _V
12	6, 7, 11	fvmpt 5956	. . 3 |- ( C e. Cat -> (Nat ` C ) = U. ran H )
13	7	dmmptss 5509	. . . . . . 7 |- dom Nat C_ Cat
14	13	sseli 3495	. . . . . 6 |- ( C e. dom Nat -> C e. Cat )
15	14	con3i 135	. . . . 5 |- ( -. C e. Cat -> -. C e. dom Nat )
16		ndmfv 5896	. . . . 5 |- ( -. C e. dom Nat -> (Nat ` C ) = (/) )
17	15, 16	syl 16	. . . 4 |- ( -. C e. Cat -> (Nat ` C ) = (/) )
18		df-homa 15432	. . . . . . . . . . . . 13 |- Homa = ( c e. Cat |-> ( x e. ( ( Base ` c ) X. ( Base ` c ) ) |-> ( { x } X. ( ( Hom ` c ) ` x ) ) ) )
19	18	dmmptss 5509	. . . . . . . . . . . 12 |- dom Homa C_ Cat
20	19	sseli 3495	. . . . . . . . . . 11 |- ( C e. dom Homa -> C e. Cat )
21	20	con3i 135	. . . . . . . . . 10 |- ( -. C e. Cat -> -. C e. dom Homa )
22		ndmfv 5896	. . . . . . . . . 10 |- ( -. C e. dom Homa -> (Homa ` C ) = (/) )
23	21, 22	syl 16	. . . . . . . . 9 |- ( -. C e. Cat -> (Homa ` C ) = (/) )
24	3, 23	syl5eq 2510	. . . . . . . 8 |- ( -. C e. Cat -> H = (/) )
25	24	rneqd 5240	. . . . . . 7 |- ( -. C e. Cat -> ran H = ran (/) )
26		rn0 5264	. . . . . . 7 |- ran (/) = (/)
27	25, 26	syl6eq 2514	. . . . . 6 |- ( -. C e. Cat -> ran H = (/) )
28	27	unieqd 4261	. . . . 5 |- ( -. C e. Cat -> U. ran H = U. (/) )
29		uni0 4278	. . . . 5 |- U. (/) = (/)
30	28, 29	syl6eq 2514	. . . 4 |- ( -. C e. Cat -> U. ran H = (/) )
31	17, 30	eqtr4d 2501	. . 3 |- ( -. C e. Cat -> (Nat ` C ) = U. ran H )
32	12, 31	pm2.61i 164	. 2 |- (Nat ` C ) = U. ran H
33	1, 32	eqtri 2486	1 |- A = U. ran H

Syntax hints:   -. wn 3   = wceq 1395   e. wcel 1819   _Vcvv 3109   (/)c0 3793   {csn 4032   U.cuni 4251   |-> cmpt 4515   X. cxp 5006   dom cdm 5008   ran crn 5009   ` cfv 5594   Basecbs 14644   Hom chom 14723   Catccat 15081  Natcarw 15428  Hom_achoma 15429

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fv 5602  df-homa 15432  df-arw 15433

This theorem is referenced by:  arwhoma  15451  homarw  15452