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Theorem arwval 17291
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 6663 . . . . . . 7 (𝑐 = 𝐶 → (Homa𝑐) = (Homa𝐶))
3 arwval.h . . . . . . 7 𝐻 = (Homa𝐶)
42, 3syl6eqr 2871 . . . . . 6 (𝑐 = 𝐶 → (Homa𝑐) = 𝐻)
54rneqd 5801 . . . . 5 (𝑐 = 𝐶 → ran (Homa𝑐) = ran 𝐻)
65unieqd 4840 . . . 4 (𝑐 = 𝐶 ran (Homa𝑐) = ran 𝐻)
7 df-arw 17275 . . . 4 Arrow = (𝑐 ∈ Cat ↦ ran (Homa𝑐))
83fvexi 6677 . . . . . 6 𝐻 ∈ V
98rnex 7606 . . . . 5 ran 𝐻 ∈ V
109uniex 7454 . . . 4 ran 𝐻 ∈ V
116, 7, 10fvmpt 6761 . . 3 (𝐶 ∈ Cat → (Arrow‘𝐶) = ran 𝐻)
127fvmptndm 6790 . . . 4 𝐶 ∈ Cat → (Arrow‘𝐶) = ∅)
13 df-homa 17274 . . . . . . . . . 10 Homa = (𝑐 ∈ Cat ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥))))
1413fvmptndm 6790 . . . . . . . . 9 𝐶 ∈ Cat → (Homa𝐶) = ∅)
153, 14syl5eq 2865 . . . . . . . 8 𝐶 ∈ Cat → 𝐻 = ∅)
1615rneqd 5801 . . . . . . 7 𝐶 ∈ Cat → ran 𝐻 = ran ∅)
17 rn0 5789 . . . . . . 7 ran ∅ = ∅
1816, 17syl6eq 2869 . . . . . 6 𝐶 ∈ Cat → ran 𝐻 = ∅)
1918unieqd 4840 . . . . 5 𝐶 ∈ Cat → ran 𝐻 = ∅)
20 uni0 4857 . . . . 5 ∅ = ∅
2119, 20syl6eq 2869 . . . 4 𝐶 ∈ Cat → ran 𝐻 = ∅)
2212, 21eqtr4d 2856 . . 3 𝐶 ∈ Cat → (Arrow‘𝐶) = ran 𝐻)
2311, 22pm2.61i 183 . 2 (Arrow‘𝐶) = ran 𝐻
241, 23eqtri 2841 1 𝐴 = ran 𝐻
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1528  wcel 2105  c0 4288  {csn 4557   cuni 4830  cmpt 5137   × cxp 5546  ran crn 5549  cfv 6348  Basecbs 16471  Hom chom 16564  Catccat 16923  Arrowcarw 17270  Homachoma 17271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fv 6356  df-homa 17274  df-arw 17275
This theorem is referenced by:  arwhoma  17293  homarw  17294
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