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Theorem arwval 16516
 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 6103 . . . . . . 7 (𝑐 = 𝐶 → (Homa𝑐) = (Homa𝐶))
3 arwval.h . . . . . . 7 𝐻 = (Homa𝐶)
42, 3syl6eqr 2662 . . . . . 6 (𝑐 = 𝐶 → (Homa𝑐) = 𝐻)
54rneqd 5274 . . . . 5 (𝑐 = 𝐶 → ran (Homa𝑐) = ran 𝐻)
65unieqd 4382 . . . 4 (𝑐 = 𝐶 ran (Homa𝑐) = ran 𝐻)
7 df-arw 16500 . . . 4 Arrow = (𝑐 ∈ Cat ↦ ran (Homa𝑐))
8 fvex 6113 . . . . . . 7 (Homa𝐶) ∈ V
93, 8eqeltri 2684 . . . . . 6 𝐻 ∈ V
109rnex 6992 . . . . 5 ran 𝐻 ∈ V
1110uniex 6851 . . . 4 ran 𝐻 ∈ V
126, 7, 11fvmpt 6191 . . 3 (𝐶 ∈ Cat → (Arrow‘𝐶) = ran 𝐻)
137dmmptss 5548 . . . . . . 7 dom Arrow ⊆ Cat
1413sseli 3564 . . . . . 6 (𝐶 ∈ dom Arrow → 𝐶 ∈ Cat)
1514con3i 149 . . . . 5 𝐶 ∈ Cat → ¬ 𝐶 ∈ dom Arrow)
16 ndmfv 6128 . . . . 5 𝐶 ∈ dom Arrow → (Arrow‘𝐶) = ∅)
1715, 16syl 17 . . . 4 𝐶 ∈ Cat → (Arrow‘𝐶) = ∅)
18 df-homa 16499 . . . . . . . . . . . . 13 Homa = (𝑐 ∈ Cat ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥))))
1918dmmptss 5548 . . . . . . . . . . . 12 dom Homa ⊆ Cat
2019sseli 3564 . . . . . . . . . . 11 (𝐶 ∈ dom Homa𝐶 ∈ Cat)
2120con3i 149 . . . . . . . . . 10 𝐶 ∈ Cat → ¬ 𝐶 ∈ dom Homa)
22 ndmfv 6128 . . . . . . . . . 10 𝐶 ∈ dom Homa → (Homa𝐶) = ∅)
2321, 22syl 17 . . . . . . . . 9 𝐶 ∈ Cat → (Homa𝐶) = ∅)
243, 23syl5eq 2656 . . . . . . . 8 𝐶 ∈ Cat → 𝐻 = ∅)
2524rneqd 5274 . . . . . . 7 𝐶 ∈ Cat → ran 𝐻 = ran ∅)
26 rn0 5298 . . . . . . 7 ran ∅ = ∅
2725, 26syl6eq 2660 . . . . . 6 𝐶 ∈ Cat → ran 𝐻 = ∅)
2827unieqd 4382 . . . . 5 𝐶 ∈ Cat → ran 𝐻 = ∅)
29 uni0 4401 . . . . 5 ∅ = ∅
3028, 29syl6eq 2660 . . . 4 𝐶 ∈ Cat → ran 𝐻 = ∅)
3117, 30eqtr4d 2647 . . 3 𝐶 ∈ Cat → (Arrow‘𝐶) = ran 𝐻)
3212, 31pm2.61i 175 . 2 (Arrow‘𝐶) = ran 𝐻
331, 32eqtri 2632 1 𝐴 = ran 𝐻
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ∅c0 3874  {csn 4125  ∪ cuni 4372   ↦ cmpt 4643   × cxp 5036  dom cdm 5038  ran crn 5039  ‘cfv 5804  Basecbs 15695  Hom chom 15779  Catccat 16148  Arrowcarw 16495  Homachoma 16496 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fv 5812  df-homa 16499  df-arw 16500 This theorem is referenced by:  arwhoma  16518  homarw  16519
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