Theorem arwval 18110

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 6920	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (Homa‘𝑐) = (Homa‘𝐶))
3		arwval.h	. . . . . . 7 ⊢ 𝐻 = (Homa‘𝐶)
4	2, 3	eqtr4di 2798	. . . . . 6 ⊢ (𝑐 = 𝐶 → (Homa‘𝑐) = 𝐻)
5	4	rneqd 5963	. . . . 5 ⊢ (𝑐 = 𝐶 → ran (Homa‘𝑐) = ran 𝐻)
6	5	unieqd 4944	. . . 4 ⊢ (𝑐 = 𝐶 → ∪ ran (Homa‘𝑐) = ∪ ran 𝐻)
7		df-arw 18094	. . . 4 ⊢ Arrow = (𝑐 ∈ Cat ↦ ∪ ran (Homa‘𝑐))
8	3	fvexi 6934	. . . . . 6 ⊢ 𝐻 ∈ V
9	8	rnex 7950	. . . . 5 ⊢ ran 𝐻 ∈ V
10	9	uniex 7776	. . . 4 ⊢ ∪ ran 𝐻 ∈ V
11	6, 7, 10	fvmpt 7029	. . 3 ⊢ (𝐶 ∈ Cat → (Arrow‘𝐶) = ∪ ran 𝐻)
12	7	fvmptndm 7060	. . . 4 ⊢ (¬ 𝐶 ∈ Cat → (Arrow‘𝐶) = ∅)
13		df-homa 18093	. . . . . . . . . 10 ⊢ Homa = (𝑐 ∈ Cat ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥))))
14	13	fvmptndm 7060	. . . . . . . . 9 ⊢ (¬ 𝐶 ∈ Cat → (Homa‘𝐶) = ∅)
15	3, 14	eqtrid 2792	. . . . . . . 8 ⊢ (¬ 𝐶 ∈ Cat → 𝐻 = ∅)
16	15	rneqd 5963	. . . . . . 7 ⊢ (¬ 𝐶 ∈ Cat → ran 𝐻 = ran ∅)
17		rn0 5950	. . . . . . 7 ⊢ ran ∅ = ∅
18	16, 17	eqtrdi 2796	. . . . . 6 ⊢ (¬ 𝐶 ∈ Cat → ran 𝐻 = ∅)
19	18	unieqd 4944	. . . . 5 ⊢ (¬ 𝐶 ∈ Cat → ∪ ran 𝐻 = ∪ ∅)
20		uni0 4959	. . . . 5 ⊢ ∪ ∅ = ∅
21	19, 20	eqtrdi 2796	. . . 4 ⊢ (¬ 𝐶 ∈ Cat → ∪ ran 𝐻 = ∅)
22	12, 21	eqtr4d 2783	. . 3 ⊢ (¬ 𝐶 ∈ Cat → (Arrow‘𝐶) = ∪ ran 𝐻)
23	11, 22	pm2.61i 182	. 2 ⊢ (Arrow‘𝐶) = ∪ ran 𝐻
24	1, 23	eqtri 2768	1 ⊢ 𝐴 = ∪ ran 𝐻

Syntax hints:  ¬ wn 3   = wceq 1537   ∈ wcel 2108  ∅c0 4352  {csn 4648  ∪ cuni 4931   ↦ cmpt 5249   × cxp 5698  ran crn 5701  ‘cfv 6573  Basecbs 17258  Hom chom 17322  Catccat 17722  Arrowcarw 18089  Hom_achoma 18090

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6525  df-fun 6575  df-fv 6581  df-homa 18093  df-arw 18094

This theorem is referenced by:  arwhoma  18112  homarw  18113