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 = (Arrow‘𝐶)
arwval.h	⊢ 𝐻 = (Homa‘𝐶)

Assertion

Ref	Expression
arwval	⊢ 𝐴 = ∪ 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 = (Arrow‘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 → ∪ ran (Homa‘𝑐) = ∪ ran 𝐻)
7		df-arw 16677	. . . 4 ⊢ Arrow = (𝑐 ∈ Cat ↦ ∪ ran (Homa‘𝑐))
8		fvex 6201	. . . . . . 7 ⊢ (Homa‘𝐶) ∈ V
9	3, 8	eqeltri 2697	. . . . . 6 ⊢ 𝐻 ∈ V
10	9	rnex 7100	. . . . 5 ⊢ ran 𝐻 ∈ V
11	10	uniex 6953	. . . 4 ⊢ ∪ ran 𝐻 ∈ V
12	6, 7, 11	fvmpt 6282	. . 3 ⊢ (𝐶 ∈ Cat → (Arrow‘𝐶) = ∪ ran 𝐻)
13	7	dmmptss 5631	. . . . . . 7 ⊢ dom Arrow ⊆ Cat
14	13	sseli 3599	. . . . . 6 ⊢ (𝐶 ∈ dom Arrow → 𝐶 ∈ Cat)
15	14	con3i 150	. . . . 5 ⊢ (¬ 𝐶 ∈ Cat → ¬ 𝐶 ∈ dom Arrow)
16		ndmfv 6218	. . . . 5 ⊢ (¬ 𝐶 ∈ dom Arrow → (Arrow‘𝐶) = ∅)
17	15, 16	syl 17	. . . 4 ⊢ (¬ 𝐶 ∈ Cat → (Arrow‘𝐶) = ∅)
18		df-homa 16676	. . . . . . . . . . . . 13 ⊢ Homa = (𝑐 ∈ Cat ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥))))
19	18	dmmptss 5631	. . . . . . . . . . . 12 ⊢ dom Homa ⊆ Cat
20	19	sseli 3599	. . . . . . . . . . 11 ⊢ (𝐶 ∈ dom Homa → 𝐶 ∈ Cat)
21	20	con3i 150	. . . . . . . . . 10 ⊢ (¬ 𝐶 ∈ Cat → ¬ 𝐶 ∈ dom Homa)
22		ndmfv 6218	. . . . . . . . . 10 ⊢ (¬ 𝐶 ∈ dom Homa → (Homa‘𝐶) = ∅)
23	21, 22	syl 17	. . . . . . . . 9 ⊢ (¬ 𝐶 ∈ Cat → (Homa‘𝐶) = ∅)
24	3, 23	syl5eq 2668	. . . . . . . 8 ⊢ (¬ 𝐶 ∈ Cat → 𝐻 = ∅)
25	24	rneqd 5353	. . . . . . 7 ⊢ (¬ 𝐶 ∈ Cat → ran 𝐻 = ran ∅)
26		rn0 5377	. . . . . . 7 ⊢ ran ∅ = ∅
27	25, 26	syl6eq 2672	. . . . . 6 ⊢ (¬ 𝐶 ∈ Cat → ran 𝐻 = ∅)
28	27	unieqd 4446	. . . . 5 ⊢ (¬ 𝐶 ∈ Cat → ∪ ran 𝐻 = ∪ ∅)
29		uni0 4465	. . . . 5 ⊢ ∪ ∅ = ∅
30	28, 29	syl6eq 2672	. . . 4 ⊢ (¬ 𝐶 ∈ Cat → ∪ ran 𝐻 = ∅)
31	17, 30	eqtr4d 2659	. . 3 ⊢ (¬ 𝐶 ∈ Cat → (Arrow‘𝐶) = ∪ ran 𝐻)
32	12, 31	pm2.61i 176	. 2 ⊢ (Arrow‘𝐶) = ∪ ran 𝐻
33	1, 32	eqtri 2644	1 ⊢ 𝐴 = ∪ ran 𝐻

Syntax hints:  ¬ wn 3   = wceq 1483   ∈ wcel 1990  Vcvv 3200  ∅c0 3915  {csn 4177  ∪ cuni 4436   ↦ cmpt 4729   × cxp 5112  dom cdm 5114  ran crn 5115  ‘cfv 5888  Basecbs 15857  Hom chom 15952  Catccat 16325  Arrowcarw 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