Theorem arwval 18198

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 6877	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (Homa‘𝑐) = (Homa‘𝐶))
3		arwval.h	. . . . . . 7 ⊢ 𝐻 = (Homa‘𝐶)
4	2, 3	eqtr4di 2814	. . . . . 6 ⊢ (𝑐 = 𝐶 → (Homa‘𝑐) = 𝐻)
5	4	rneqd 5920	. . . . 5 ⊢ (𝑐 = 𝐶 → ran (Homa‘𝑐) = ran 𝐻)
6	5	unieqd 4880	. . . 4 ⊢ (𝑐 = 𝐶 → ∪ ran (Homa‘𝑐) = ∪ ran 𝐻)
7		df-arw 18182	. . . 4 ⊢ Arrow = (𝑐 ∈ Cat ↦ ∪ ran (Homa‘𝑐))
8	3	fvexi 6891	. . . . . 6 ⊢ 𝐻 ∈ V
9	8	rnex 7911	. . . . 5 ⊢ ran 𝐻 ∈ V
10	9	uniex 7747	. . . 4 ⊢ ∪ ran 𝐻 ∈ V
11	6, 7, 10	fvmpt 6985	. . 3 ⊢ (𝐶 ∈ Cat → (Arrow‘𝐶) = ∪ ran 𝐻)
12	7	fvmptndm 7017	. . . 4 ⊢ (¬ 𝐶 ∈ Cat → (Arrow‘𝐶) = ∅)
13		df-homa 18181	. . . . . . . . . 10 ⊢ Homa = (𝑐 ∈ Cat ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥))))
14	13	fvmptndm 7017	. . . . . . . . 9 ⊢ (¬ 𝐶 ∈ Cat → (Homa‘𝐶) = ∅)
15	3, 14	eqtrid 2808	. . . . . . . 8 ⊢ (¬ 𝐶 ∈ Cat → 𝐻 = ∅)
16	15	rneqd 5920	. . . . . . 7 ⊢ (¬ 𝐶 ∈ Cat → ran 𝐻 = ran ∅)
17		rn0 5908	. . . . . . 7 ⊢ ran ∅ = ∅
18	16, 17	eqtrdi 2812	. . . . . 6 ⊢ (¬ 𝐶 ∈ Cat → ran 𝐻 = ∅)
19	18	unieqd 4880	. . . . 5 ⊢ (¬ 𝐶 ∈ Cat → ∪ ran 𝐻 = ∪ ∅)
20		uni0 4896	. . . . 5 ⊢ ∪ ∅ = ∅
21	19, 20	eqtrdi 2812	. . . 4 ⊢ (¬ 𝐶 ∈ Cat → ∪ ran 𝐻 = ∅)
22	12, 21	eqtr4d 2799	. . 3 ⊢ (¬ 𝐶 ∈ Cat → (Arrow‘𝐶) = ∪ ran 𝐻)
23	11, 22	pm2.61i 184	. 2 ⊢ (Arrow‘𝐶) = ∪ ran 𝐻
24	1, 23	eqtri 2784	1 ⊢ 𝐴 = ∪ ran 𝐻

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570   ∈ wcel 2145  ∅c0 4279  {csn 4584  ∪ cuni 4867   ↦ cmpt 5186   × cxp 5649  ran crn 5652  ‘cfv 6531  Basecbs 17367  Hom chom 17419  Catccat 17818  Arrowcarw 18177  Hom_achoma 18178

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7740

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 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-iota 6487  df-fun 6533  df-fv 6539  df-homa 18181  df-arw 18182

This theorem is used by:  arwhoma  18200  homarw  18201