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Theorem arwval 18097
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 6907 . . . . . . 7 (𝑐 = 𝐶 → (Homa𝑐) = (Homa𝐶))
3 arwval.h . . . . . . 7 𝐻 = (Homa𝐶)
42, 3eqtr4di 2793 . . . . . 6 (𝑐 = 𝐶 → (Homa𝑐) = 𝐻)
54rneqd 5952 . . . . 5 (𝑐 = 𝐶 → ran (Homa𝑐) = ran 𝐻)
65unieqd 4925 . . . 4 (𝑐 = 𝐶 ran (Homa𝑐) = ran 𝐻)
7 df-arw 18081 . . . 4 Arrow = (𝑐 ∈ Cat ↦ ran (Homa𝑐))
83fvexi 6921 . . . . . 6 𝐻 ∈ V
98rnex 7933 . . . . 5 ran 𝐻 ∈ V
109uniex 7760 . . . 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 18080 . . . . . . . . . 10 Homa = (𝑐 ∈ Cat ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥))))
1413fvmptndm 7047 . . . . . . . . 9 𝐶 ∈ Cat → (Homa𝐶) = ∅)
153, 14eqtrid 2787 . . . . . . . 8 𝐶 ∈ Cat → 𝐻 = ∅)
1615rneqd 5952 . . . . . . 7 𝐶 ∈ Cat → ran 𝐻 = ran ∅)
17 rn0 5939 . . . . . . 7 ran ∅ = ∅
1816, 17eqtrdi 2791 . . . . . 6 𝐶 ∈ Cat → ran 𝐻 = ∅)
1918unieqd 4925 . . . . 5 𝐶 ∈ Cat → ran 𝐻 = ∅)
20 uni0 4940 . . . . 5 ∅ = ∅
2119, 20eqtrdi 2791 . . . 4 𝐶 ∈ Cat → ran 𝐻 = ∅)
2212, 21eqtr4d 2778 . . 3 𝐶 ∈ Cat → (Arrow‘𝐶) = ran 𝐻)
2311, 22pm2.61i 182 . 2 (Arrow‘𝐶) = ran 𝐻
241, 23eqtri 2763 1 𝐴 = ran 𝐻
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1537  wcel 2106  c0 4339  {csn 4631   cuni 4912  cmpt 5231   × cxp 5687  ran crn 5690  cfv 6563  Basecbs 17245  Hom chom 17309  Catccat 17709  Arrowcarw 18076  Homachoma 18077
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fv 6571  df-homa 18080  df-arw 18081
This theorem is referenced by:  arwhoma  18099  homarw  18100
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