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Theorem arwval 17298
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 6669 . . . . . . 7 (𝑐 = 𝐶 → (Homa𝑐) = (Homa𝐶))
3 arwval.h . . . . . . 7 𝐻 = (Homa𝐶)
42, 3syl6eqr 2879 . . . . . 6 (𝑐 = 𝐶 → (Homa𝑐) = 𝐻)
54rneqd 5807 . . . . 5 (𝑐 = 𝐶 → ran (Homa𝑐) = ran 𝐻)
65unieqd 4847 . . . 4 (𝑐 = 𝐶 ran (Homa𝑐) = ran 𝐻)
7 df-arw 17282 . . . 4 Arrow = (𝑐 ∈ Cat ↦ ran (Homa𝑐))
83fvexi 6683 . . . . . 6 𝐻 ∈ V
98rnex 7610 . . . . 5 ran 𝐻 ∈ V
109uniex 7459 . . . 4 ran 𝐻 ∈ V
116, 7, 10fvmpt 6767 . . 3 (𝐶 ∈ Cat → (Arrow‘𝐶) = ran 𝐻)
127fvmptndm 6796 . . . 4 𝐶 ∈ Cat → (Arrow‘𝐶) = ∅)
13 df-homa 17281 . . . . . . . . . 10 Homa = (𝑐 ∈ Cat ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥))))
1413fvmptndm 6796 . . . . . . . . 9 𝐶 ∈ Cat → (Homa𝐶) = ∅)
153, 14syl5eq 2873 . . . . . . . 8 𝐶 ∈ Cat → 𝐻 = ∅)
1615rneqd 5807 . . . . . . 7 𝐶 ∈ Cat → ran 𝐻 = ran ∅)
17 rn0 5795 . . . . . . 7 ran ∅ = ∅
1816, 17syl6eq 2877 . . . . . 6 𝐶 ∈ Cat → ran 𝐻 = ∅)
1918unieqd 4847 . . . . 5 𝐶 ∈ Cat → ran 𝐻 = ∅)
20 uni0 4864 . . . . 5 ∅ = ∅
2119, 20syl6eq 2877 . . . 4 𝐶 ∈ Cat → ran 𝐻 = ∅)
2212, 21eqtr4d 2864 . . 3 𝐶 ∈ Cat → (Arrow‘𝐶) = ran 𝐻)
2311, 22pm2.61i 183 . 2 (Arrow‘𝐶) = ran 𝐻
241, 23eqtri 2849 1 𝐴 = ran 𝐻
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1530  wcel 2107  c0 4295  {csn 4564   cuni 4837  cmpt 5143   × cxp 5552  ran crn 5555  cfv 6354  Basecbs 16478  Hom chom 16571  Catccat 16930  Arrowcarw 17277  Homachoma 17278
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-iota 6313  df-fun 6356  df-fv 6362  df-homa 17281  df-arw 17282
This theorem is referenced by:  arwhoma  17300  homarw  17301
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