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Theorem arwval 16740
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 6229 . . . . . . 7 (𝑐 = 𝐶 → (Homa𝑐) = (Homa𝐶))
3 arwval.h . . . . . . 7 𝐻 = (Homa𝐶)
42, 3syl6eqr 2703 . . . . . 6 (𝑐 = 𝐶 → (Homa𝑐) = 𝐻)
54rneqd 5385 . . . . 5 (𝑐 = 𝐶 → ran (Homa𝑐) = ran 𝐻)
65unieqd 4478 . . . 4 (𝑐 = 𝐶 ran (Homa𝑐) = ran 𝐻)
7 df-arw 16724 . . . 4 Arrow = (𝑐 ∈ Cat ↦ ran (Homa𝑐))
8 fvex 6239 . . . . . . 7 (Homa𝐶) ∈ V
93, 8eqeltri 2726 . . . . . 6 𝐻 ∈ V
109rnex 7142 . . . . 5 ran 𝐻 ∈ V
1110uniex 6995 . . . 4 ran 𝐻 ∈ V
126, 7, 11fvmpt 6321 . . 3 (𝐶 ∈ Cat → (Arrow‘𝐶) = ran 𝐻)
137dmmptss 5669 . . . . . . 7 dom Arrow ⊆ Cat
1413sseli 3632 . . . . . 6 (𝐶 ∈ dom Arrow → 𝐶 ∈ Cat)
1514con3i 150 . . . . 5 𝐶 ∈ Cat → ¬ 𝐶 ∈ dom Arrow)
16 ndmfv 6256 . . . . 5 𝐶 ∈ dom Arrow → (Arrow‘𝐶) = ∅)
1715, 16syl 17 . . . 4 𝐶 ∈ Cat → (Arrow‘𝐶) = ∅)
18 df-homa 16723 . . . . . . . . . . . . 13 Homa = (𝑐 ∈ Cat ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥))))
1918dmmptss 5669 . . . . . . . . . . . 12 dom Homa ⊆ Cat
2019sseli 3632 . . . . . . . . . . 11 (𝐶 ∈ dom Homa𝐶 ∈ Cat)
2120con3i 150 . . . . . . . . . 10 𝐶 ∈ Cat → ¬ 𝐶 ∈ dom Homa)
22 ndmfv 6256 . . . . . . . . . 10 𝐶 ∈ dom Homa → (Homa𝐶) = ∅)
2321, 22syl 17 . . . . . . . . 9 𝐶 ∈ Cat → (Homa𝐶) = ∅)
243, 23syl5eq 2697 . . . . . . . 8 𝐶 ∈ Cat → 𝐻 = ∅)
2524rneqd 5385 . . . . . . 7 𝐶 ∈ Cat → ran 𝐻 = ran ∅)
26 rn0 5409 . . . . . . 7 ran ∅ = ∅
2725, 26syl6eq 2701 . . . . . 6 𝐶 ∈ Cat → ran 𝐻 = ∅)
2827unieqd 4478 . . . . 5 𝐶 ∈ Cat → ran 𝐻 = ∅)
29 uni0 4497 . . . . 5 ∅ = ∅
3028, 29syl6eq 2701 . . . 4 𝐶 ∈ Cat → ran 𝐻 = ∅)
3117, 30eqtr4d 2688 . . 3 𝐶 ∈ Cat → (Arrow‘𝐶) = ran 𝐻)
3212, 31pm2.61i 176 . 2 (Arrow‘𝐶) = ran 𝐻
331, 32eqtri 2673 1 𝐴 = ran 𝐻
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1523  wcel 2030  Vcvv 3231  c0 3948  {csn 4210   cuni 4468  cmpt 4762   × cxp 5141  dom cdm 5143  ran crn 5144  cfv 5926  Basecbs 15904  Hom chom 15999  Catccat 16372  Arrowcarw 16719  Homachoma 16720
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fv 5934  df-homa 16723  df-arw 16724
This theorem is referenced by:  arwhoma  16742  homarw  16743
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