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Theorem arwval 18088
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 = (Homa𝐶)
Assertion
Ref Expression
arwval 𝐴 = ran 𝐻

Proof of Theorem arwval
Dummy variables 𝑥 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 arwval.a . 2 𝐴 = (Arrow‘𝐶)
2 fveq2 6906 . . . . . . 7 (𝑐 = 𝐶 → (Homa𝑐) = (Homa𝐶))
3 arwval.h . . . . . . 7 𝐻 = (Homa𝐶)
42, 3eqtr4di 2795 . . . . . 6 (𝑐 = 𝐶 → (Homa𝑐) = 𝐻)
54rneqd 5949 . . . . 5 (𝑐 = 𝐶 → ran (Homa𝑐) = ran 𝐻)
65unieqd 4920 . . . 4 (𝑐 = 𝐶 ran (Homa𝑐) = ran 𝐻)
7 df-arw 18072 . . . 4 Arrow = (𝑐 ∈ Cat ↦ ran (Homa𝑐))
83fvexi 6920 . . . . . 6 𝐻 ∈ V
98rnex 7932 . . . . 5 ran 𝐻 ∈ V
109uniex 7761 . . . 4 ran 𝐻 ∈ V
116, 7, 10fvmpt 7016 . . 3 (𝐶 ∈ Cat → (Arrow‘𝐶) = ran 𝐻)
127fvmptndm 7047 . . . 4 𝐶 ∈ Cat → (Arrow‘𝐶) = ∅)
13 df-homa 18071 . . . . . . . . . 10 Homa = (𝑐 ∈ Cat ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥))))
1413fvmptndm 7047 . . . . . . . . 9 𝐶 ∈ Cat → (Homa𝐶) = ∅)
153, 14eqtrid 2789 . . . . . . . 8 𝐶 ∈ Cat → 𝐻 = ∅)
1615rneqd 5949 . . . . . . 7 𝐶 ∈ Cat → ran 𝐻 = ran ∅)
17 rn0 5936 . . . . . . 7 ran ∅ = ∅
1816, 17eqtrdi 2793 . . . . . 6 𝐶 ∈ Cat → ran 𝐻 = ∅)
1918unieqd 4920 . . . . 5 𝐶 ∈ Cat → ran 𝐻 = ∅)
20 uni0 4935 . . . . 5 ∅ = ∅
2119, 20eqtrdi 2793 . . . 4 𝐶 ∈ Cat → ran 𝐻 = ∅)
2212, 21eqtr4d 2780 . . 3 𝐶 ∈ Cat → (Arrow‘𝐶) = ran 𝐻)
2311, 22pm2.61i 182 . 2 (Arrow‘𝐶) = ran 𝐻
241, 23eqtri 2765 1 𝐴 = ran 𝐻
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1540  wcel 2108  c0 4333  {csn 4626   cuni 4907  cmpt 5225   × cxp 5683  ran crn 5686  cfv 6561  Basecbs 17247  Hom chom 17308  Catccat 17707  Arrowcarw 18067  Homachoma 18068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fv 6569  df-homa 18071  df-arw 18072
This theorem is referenced by:  arwhoma  18090  homarw  18091
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