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Theorem arwval 17993
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 = (Arrowβ€˜πΆ)
arwval.h 𝐻 = (Homaβ€˜πΆ)
Assertion
Ref Expression
arwval 𝐴 = βˆͺ ran 𝐻

Proof of Theorem arwval
Dummy variables π‘₯ 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 arwval.a . 2 𝐴 = (Arrowβ€˜πΆ)
2 fveq2 6892 . . . . . . 7 (𝑐 = 𝐢 β†’ (Homaβ€˜π‘) = (Homaβ€˜πΆ))
3 arwval.h . . . . . . 7 𝐻 = (Homaβ€˜πΆ)
42, 3eqtr4di 2791 . . . . . 6 (𝑐 = 𝐢 β†’ (Homaβ€˜π‘) = 𝐻)
54rneqd 5938 . . . . 5 (𝑐 = 𝐢 β†’ ran (Homaβ€˜π‘) = ran 𝐻)
65unieqd 4923 . . . 4 (𝑐 = 𝐢 β†’ βˆͺ ran (Homaβ€˜π‘) = βˆͺ ran 𝐻)
7 df-arw 17977 . . . 4 Arrow = (𝑐 ∈ Cat ↦ βˆͺ ran (Homaβ€˜π‘))
83fvexi 6906 . . . . . 6 𝐻 ∈ V
98rnex 7903 . . . . 5 ran 𝐻 ∈ V
109uniex 7731 . . . 4 βˆͺ ran 𝐻 ∈ V
116, 7, 10fvmpt 6999 . . 3 (𝐢 ∈ Cat β†’ (Arrowβ€˜πΆ) = βˆͺ ran 𝐻)
127fvmptndm 7029 . . . 4 (Β¬ 𝐢 ∈ Cat β†’ (Arrowβ€˜πΆ) = βˆ…)
13 df-homa 17976 . . . . . . . . . 10 Homa = (𝑐 ∈ Cat ↦ (π‘₯ ∈ ((Baseβ€˜π‘) Γ— (Baseβ€˜π‘)) ↦ ({π‘₯} Γ— ((Hom β€˜π‘)β€˜π‘₯))))
1413fvmptndm 7029 . . . . . . . . 9 (Β¬ 𝐢 ∈ Cat β†’ (Homaβ€˜πΆ) = βˆ…)
153, 14eqtrid 2785 . . . . . . . 8 (Β¬ 𝐢 ∈ Cat β†’ 𝐻 = βˆ…)
1615rneqd 5938 . . . . . . 7 (Β¬ 𝐢 ∈ Cat β†’ ran 𝐻 = ran βˆ…)
17 rn0 5926 . . . . . . 7 ran βˆ… = βˆ…
1816, 17eqtrdi 2789 . . . . . 6 (Β¬ 𝐢 ∈ Cat β†’ ran 𝐻 = βˆ…)
1918unieqd 4923 . . . . 5 (Β¬ 𝐢 ∈ Cat β†’ βˆͺ ran 𝐻 = βˆͺ βˆ…)
20 uni0 4940 . . . . 5 βˆͺ βˆ… = βˆ…
2119, 20eqtrdi 2789 . . . 4 (Β¬ 𝐢 ∈ Cat β†’ βˆͺ ran 𝐻 = βˆ…)
2212, 21eqtr4d 2776 . . 3 (Β¬ 𝐢 ∈ Cat β†’ (Arrowβ€˜πΆ) = βˆͺ ran 𝐻)
2311, 22pm2.61i 182 . 2 (Arrowβ€˜πΆ) = βˆͺ ran 𝐻
241, 23eqtri 2761 1 𝐴 = βˆͺ ran 𝐻
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   = wceq 1542   ∈ wcel 2107  βˆ…c0 4323  {csn 4629  βˆͺ cuni 4909   ↦ cmpt 5232   Γ— cxp 5675  ran crn 5678  β€˜cfv 6544  Basecbs 17144  Hom chom 17208  Catccat 17608  Arrowcarw 17972  Homachoma 17973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fv 6552  df-homa 17976  df-arw 17977
This theorem is referenced by:  arwhoma  17995  homarw  17996
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