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Theorem arwval 17975
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 6878 . . . . . . 7 (𝑐 = 𝐶 → (Homa𝑐) = (Homa𝐶))
3 arwval.h . . . . . . 7 𝐻 = (Homa𝐶)
42, 3eqtr4di 2789 . . . . . 6 (𝑐 = 𝐶 → (Homa𝑐) = 𝐻)
54rneqd 5929 . . . . 5 (𝑐 = 𝐶 → ran (Homa𝑐) = ran 𝐻)
65unieqd 4915 . . . 4 (𝑐 = 𝐶 ran (Homa𝑐) = ran 𝐻)
7 df-arw 17959 . . . 4 Arrow = (𝑐 ∈ Cat ↦ ran (Homa𝑐))
83fvexi 6892 . . . . . 6 𝐻 ∈ V
98rnex 7885 . . . . 5 ran 𝐻 ∈ V
109uniex 7714 . . . 4 ran 𝐻 ∈ V
116, 7, 10fvmpt 6984 . . 3 (𝐶 ∈ Cat → (Arrow‘𝐶) = ran 𝐻)
127fvmptndm 7014 . . . 4 𝐶 ∈ Cat → (Arrow‘𝐶) = ∅)
13 df-homa 17958 . . . . . . . . . 10 Homa = (𝑐 ∈ Cat ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥))))
1413fvmptndm 7014 . . . . . . . . 9 𝐶 ∈ Cat → (Homa𝐶) = ∅)
153, 14eqtrid 2783 . . . . . . . 8 𝐶 ∈ Cat → 𝐻 = ∅)
1615rneqd 5929 . . . . . . 7 𝐶 ∈ Cat → ran 𝐻 = ran ∅)
17 rn0 5917 . . . . . . 7 ran ∅ = ∅
1816, 17eqtrdi 2787 . . . . . 6 𝐶 ∈ Cat → ran 𝐻 = ∅)
1918unieqd 4915 . . . . 5 𝐶 ∈ Cat → ran 𝐻 = ∅)
20 uni0 4932 . . . . 5 ∅ = ∅
2119, 20eqtrdi 2787 . . . 4 𝐶 ∈ Cat → ran 𝐻 = ∅)
2212, 21eqtr4d 2774 . . 3 𝐶 ∈ Cat → (Arrow‘𝐶) = ran 𝐻)
2311, 22pm2.61i 182 . 2 (Arrow‘𝐶) = ran 𝐻
241, 23eqtri 2759 1 𝐴 = ran 𝐻
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1541  wcel 2106  c0 4318  {csn 4622   cuni 4901  cmpt 5224   × cxp 5667  ran crn 5670  cfv 6532  Basecbs 17126  Hom chom 17190  Catccat 17590  Arrowcarw 17954  Homachoma 17955
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7708
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6484  df-fun 6534  df-fv 6540  df-homa 17958  df-arw 17959
This theorem is referenced by:  arwhoma  17977  homarw  17978
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