Theorem arwval 17952

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 6828	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (Homa‘𝑐) = (Homa‘𝐶))
3		arwval.h	. . . . . . 7 ⊢ 𝐻 = (Homa‘𝐶)
4	2, 3	eqtr4di 2786	. . . . . 6 ⊢ (𝑐 = 𝐶 → (Homa‘𝑐) = 𝐻)
5	4	rneqd 5882	. . . . 5 ⊢ (𝑐 = 𝐶 → ran (Homa‘𝑐) = ran 𝐻)
6	5	unieqd 4871	. . . 4 ⊢ (𝑐 = 𝐶 → ∪ ran (Homa‘𝑐) = ∪ ran 𝐻)
7		df-arw 17936	. . . 4 ⊢ Arrow = (𝑐 ∈ Cat ↦ ∪ ran (Homa‘𝑐))
8	3	fvexi 6842	. . . . . 6 ⊢ 𝐻 ∈ V
9	8	rnex 7846	. . . . 5 ⊢ ran 𝐻 ∈ V
10	9	uniex 7680	. . . 4 ⊢ ∪ ran 𝐻 ∈ V
11	6, 7, 10	fvmpt 6935	. . 3 ⊢ (𝐶 ∈ Cat → (Arrow‘𝐶) = ∪ ran 𝐻)
12	7	fvmptndm 6966	. . . 4 ⊢ (¬ 𝐶 ∈ Cat → (Arrow‘𝐶) = ∅)
13		df-homa 17935	. . . . . . . . . 10 ⊢ Homa = (𝑐 ∈ Cat ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥))))
14	13	fvmptndm 6966	. . . . . . . . 9 ⊢ (¬ 𝐶 ∈ Cat → (Homa‘𝐶) = ∅)
15	3, 14	eqtrid 2780	. . . . . . . 8 ⊢ (¬ 𝐶 ∈ Cat → 𝐻 = ∅)
16	15	rneqd 5882	. . . . . . 7 ⊢ (¬ 𝐶 ∈ Cat → ran 𝐻 = ran ∅)
17		rn0 5870	. . . . . . 7 ⊢ ran ∅ = ∅
18	16, 17	eqtrdi 2784	. . . . . 6 ⊢ (¬ 𝐶 ∈ Cat → ran 𝐻 = ∅)
19	18	unieqd 4871	. . . . 5 ⊢ (¬ 𝐶 ∈ Cat → ∪ ran 𝐻 = ∪ ∅)
20		uni0 4886	. . . . 5 ⊢ ∪ ∅ = ∅
21	19, 20	eqtrdi 2784	. . . 4 ⊢ (¬ 𝐶 ∈ Cat → ∪ ran 𝐻 = ∅)
22	12, 21	eqtr4d 2771	. . 3 ⊢ (¬ 𝐶 ∈ Cat → (Arrow‘𝐶) = ∪ ran 𝐻)
23	11, 22	pm2.61i 182	. 2 ⊢ (Arrow‘𝐶) = ∪ ran 𝐻
24	1, 23	eqtri 2756	1 ⊢ 𝐴 = ∪ ran 𝐻

Syntax hints:  ¬ wn 3   = wceq 1541   ∈ wcel 2113  ∅c0 4282  {csn 4575  ∪ cuni 4858   ↦ cmpt 5174   × cxp 5617  ran crn 5620  ‘cfv 6486  Basecbs 17122  Hom chom 17174  Catccat 17572  Arrowcarw 17931  Hom_achoma 17932

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-iota 6442  df-fun 6488  df-fv 6494  df-homa 17935  df-arw 17936

This theorem is referenced by:  arwhoma  17954  homarw  17955