Theorem arwval 15950

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 5870	. . . . . . 7 |- ( c = C -> (Homa ` c ) = (Homa ` C ) )
3		arwval.h	. . . . . . 7 |- H = (Homa ` C )
4	2, 3	syl6eqr 2505	. . . . . 6 |- ( c = C -> (Homa ` c ) = H )
5	4	rneqd 5065	. . . . 5 |- ( c = C -> ran (Homa ` c ) = ran H )
6	5	unieqd 4211	. . . 4 |- ( c = C -> U. ran (Homa ` c ) = U. ran H )
7		df-arw 15934	. . . 4 |- Nat = ( c e. Cat |-> U. ran (Homa ` c ) )
8		fvex 5880	. . . . . . 7 |- (Homa ` C ) e. _V
9	3, 8	eqeltri 2527	. . . . . 6 |- H e. _V
10	9	rnex 6732	. . . . 5 |- ran H e. _V
11	10	uniex 6592	. . . 4 |- U. ran H e. _V
12	6, 7, 11	fvmpt 5953	. . 3 |- ( C e. Cat -> (Nat ` C ) = U. ran H )
13	7	dmmptss 5334	. . . . . . 7 |- dom Nat C_ Cat
14	13	sseli 3430	. . . . . 6 |- ( C e. dom Nat -> C e. Cat )
15	14	con3i 141	. . . . 5 |- ( -. C e. Cat -> -. C e. dom Nat )
16		ndmfv 5894	. . . . 5 |- ( -. C e. dom Nat -> (Nat ` C ) = (/) )
17	15, 16	syl 17	. . . 4 |- ( -. C e. Cat -> (Nat ` C ) = (/) )
18		df-homa 15933	. . . . . . . . . . . . 13 |- Homa = ( c e. Cat |-> ( x e. ( ( Base ` c ) X. ( Base ` c ) ) |-> ( { x } X. ( ( Hom ` c ) ` x ) ) ) )
19	18	dmmptss 5334	. . . . . . . . . . . 12 |- dom Homa C_ Cat
20	19	sseli 3430	. . . . . . . . . . 11 |- ( C e. dom Homa -> C e. Cat )
21	20	con3i 141	. . . . . . . . . 10 |- ( -. C e. Cat -> -. C e. dom Homa )
22		ndmfv 5894	. . . . . . . . . 10 |- ( -. C e. dom Homa -> (Homa ` C ) = (/) )
23	21, 22	syl 17	. . . . . . . . 9 |- ( -. C e. Cat -> (Homa ` C ) = (/) )
24	3, 23	syl5eq 2499	. . . . . . . 8 |- ( -. C e. Cat -> H = (/) )
25	24	rneqd 5065	. . . . . . 7 |- ( -. C e. Cat -> ran H = ran (/) )
26		rn0 5089	. . . . . . 7 |- ran (/) = (/)
27	25, 26	syl6eq 2503	. . . . . 6 |- ( -. C e. Cat -> ran H = (/) )
28	27	unieqd 4211	. . . . 5 |- ( -. C e. Cat -> U. ran H = U. (/) )
29		uni0 4228	. . . . 5 |- U. (/) = (/)
30	28, 29	syl6eq 2503	. . . 4 |- ( -. C e. Cat -> U. ran H = (/) )
31	17, 30	eqtr4d 2490	. . 3 |- ( -. C e. Cat -> (Nat ` C ) = U. ran H )
32	12, 31	pm2.61i 168	. 2 |- (Nat ` C ) = U. ran H
33	1, 32	eqtri 2475	1 |- A = U. ran H

Syntax hints:   -. wn 3   = wceq 1446   e. wcel 1889   _Vcvv 3047   (/)c0 3733   {csn 3970   U.cuni 4201   |-> cmpt 4464   X. cxp 4835   dom cdm 4837   ran crn 4838   ` cfv 5585   Basecbs 15133   Hom chom 15213   Catccat 15582  Natcarw 15929  Hom_achoma 15930

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-sbc 3270  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5549  df-fun 5587  df-fv 5593  df-homa 15933  df-arw 15934

This theorem is referenced by:  arwhoma  15952  homarw  15953