Theorem arwval 16693

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 6191	. . . . . . 7 |- ( c = C -> (Homa ` c ) = (Homa ` C ) )
3		arwval.h	. . . . . . 7 |- H = (Homa ` C )
4	2, 3	syl6eqr 2674	. . . . . 6 |- ( c = C -> (Homa ` c ) = H )
5	4	rneqd 5353	. . . . 5 |- ( c = C -> ran (Homa ` c ) = ran H )
6	5	unieqd 4446	. . . 4 |- ( c = C -> U. ran (Homa ` c ) = U. ran H )
7		df-arw 16677	. . . 4 |- Nat = ( c e. Cat |-> U. ran (Homa ` c ) )
8		fvex 6201	. . . . . . 7 |- (Homa ` C ) e. _V
9	3, 8	eqeltri 2697	. . . . . 6 |- H e. _V
10	9	rnex 7100	. . . . 5 |- ran H e. _V
11	10	uniex 6953	. . . 4 |- U. ran H e. _V
12	6, 7, 11	fvmpt 6282	. . 3 |- ( C e. Cat -> (Nat ` C ) = U. ran H )
13	7	dmmptss 5631	. . . . . . 7 |- dom Nat C_ Cat
14	13	sseli 3599	. . . . . 6 |- ( C e. dom Nat -> C e. Cat )
15	14	con3i 150	. . . . 5 |- ( -. C e. Cat -> -. C e. dom Nat )
16		ndmfv 6218	. . . . 5 |- ( -. C e. dom Nat -> (Nat ` C ) = (/) )
17	15, 16	syl 17	. . . 4 |- ( -. C e. Cat -> (Nat ` C ) = (/) )
18		df-homa 16676	. . . . . . . . . . . . 13 |- Homa = ( c e. Cat |-> ( x e. ( ( Base ` c ) X. ( Base ` c ) ) |-> ( { x } X. ( ( Hom ` c ) ` x ) ) ) )
19	18	dmmptss 5631	. . . . . . . . . . . 12 |- dom Homa C_ Cat
20	19	sseli 3599	. . . . . . . . . . 11 |- ( C e. dom Homa -> C e. Cat )
21	20	con3i 150	. . . . . . . . . 10 |- ( -. C e. Cat -> -. C e. dom Homa )
22		ndmfv 6218	. . . . . . . . . 10 |- ( -. C e. dom Homa -> (Homa ` C ) = (/) )
23	21, 22	syl 17	. . . . . . . . 9 |- ( -. C e. Cat -> (Homa ` C ) = (/) )
24	3, 23	syl5eq 2668	. . . . . . . 8 |- ( -. C e. Cat -> H = (/) )
25	24	rneqd 5353	. . . . . . 7 |- ( -. C e. Cat -> ran H = ran (/) )
26		rn0 5377	. . . . . . 7 |- ran (/) = (/)
27	25, 26	syl6eq 2672	. . . . . 6 |- ( -. C e. Cat -> ran H = (/) )
28	27	unieqd 4446	. . . . 5 |- ( -. C e. Cat -> U. ran H = U. (/) )
29		uni0 4465	. . . . 5 |- U. (/) = (/)
30	28, 29	syl6eq 2672	. . . 4 |- ( -. C e. Cat -> U. ran H = (/) )
31	17, 30	eqtr4d 2659	. . 3 |- ( -. C e. Cat -> (Nat ` C ) = U. ran H )
32	12, 31	pm2.61i 176	. 2 |- (Nat ` C ) = U. ran H
33	1, 32	eqtri 2644	1 |- A = U. ran H

Syntax hints:   -. wn 3   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   {csn 4177   U.cuni 4436   |-> cmpt 4729   X. cxp 5112   dom cdm 5114   ran crn 5115   ` cfv 5888   Basecbs 15857   Hom chom 15952   Catccat 16325  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-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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  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-fv 5896  df-homa 16676  df-arw 16677

This theorem is referenced by:  arwhoma  16695  homarw  16696