Theorem arwval 18247

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 6908	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (Homa‘𝑐) = (Homa‘𝐶))
3		arwval.h	. . . . . . 7 ⊢ 𝐻 = (Homa‘𝐶)
4	2, 3	eqtr4di 2816	. . . . . 6 ⊢ (𝑐 = 𝐶 → (Homa‘𝑐) = 𝐻)
5	4	rneqd 5927	. . . . 5 ⊢ (𝑐 = 𝐶 → ran (Homa‘𝑐) = ran 𝐻)
6	5	unieqd 4883	. . . 4 ⊢ (𝑐 = 𝐶 → ∪ ran (Homa‘𝑐) = ∪ ran 𝐻)
7		df-arw 18231	. . . 4 ⊢ Arrow = (𝑐 ∈ Cat ↦ ∪ ran (Homa‘𝑐))
8	3	fvexi 6922	. . . . . 6 ⊢ 𝐻 ∈ V
9	8	rnex 7946	. . . . 5 ⊢ ran 𝐻 ∈ V
10	9	uniex 7782	. . . 4 ⊢ ∪ ran 𝐻 ∈ V
11	6, 7, 10	fvmpt 7016	. . 3 ⊢ (𝐶 ∈ Cat → (Arrow‘𝐶) = ∪ ran 𝐻)
12	7	fvmptndm 7048	. . . 4 ⊢ (¬ 𝐶 ∈ Cat → (Arrow‘𝐶) = ∅)
13		df-homa 18230	. . . . . . . . . 10 ⊢ Homa = (𝑐 ∈ Cat ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥))))
14	13	fvmptndm 7048	. . . . . . . . 9 ⊢ (¬ 𝐶 ∈ Cat → (Homa‘𝐶) = ∅)
15	3, 14	eqtrid 2810	. . . . . . . 8 ⊢ (¬ 𝐶 ∈ Cat → 𝐻 = ∅)
16	15	rneqd 5927	. . . . . . 7 ⊢ (¬ 𝐶 ∈ Cat → ran 𝐻 = ran ∅)
17		rn0 5915	. . . . . . 7 ⊢ ran ∅ = ∅
18	16, 17	eqtrdi 2814	. . . . . 6 ⊢ (¬ 𝐶 ∈ Cat → ran 𝐻 = ∅)
19	18	unieqd 4883	. . . . 5 ⊢ (¬ 𝐶 ∈ Cat → ∪ ran 𝐻 = ∪ ∅)
20		uni0 4899	. . . . 5 ⊢ ∪ ∅ = ∅
21	19, 20	eqtrdi 2814	. . . 4 ⊢ (¬ 𝐶 ∈ Cat → ∪ ran 𝐻 = ∅)
22	12, 21	eqtr4d 2801	. . 3 ⊢ (¬ 𝐶 ∈ Cat → (Arrow‘𝐶) = ∪ ran 𝐻)
23	11, 22	pm2.61i 184	. 2 ⊢ (Arrow‘𝐶) = ∪ ran 𝐻
24	1, 23	eqtri 2786	1 ⊢ 𝐴 = ∪ ran 𝐻

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570   ∈ wcel 2147  ∅c0 4281  {csn 4588  ∪ cuni 4870   ↦ cmpt 5189   × cxp 5653  ran crn 5656  ‘cfv 6558  Basecbs 17416  Hom chom 17468  Catccat 17867  Arrowcarw 18226  Hom_achoma 18227

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6512  df-fun 6560  df-fv 6566  df-homa 18230  df-arw 18231

This theorem is used by:  arwhoma  18249  homarw  18250