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Theorem arwval 16958
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 6375 . . . . . . 7 (𝑐 = 𝐶 → (Homa𝑐) = (Homa𝐶))
3 arwval.h . . . . . . 7 𝐻 = (Homa𝐶)
42, 3syl6eqr 2817 . . . . . 6 (𝑐 = 𝐶 → (Homa𝑐) = 𝐻)
54rneqd 5521 . . . . 5 (𝑐 = 𝐶 → ran (Homa𝑐) = ran 𝐻)
65unieqd 4604 . . . 4 (𝑐 = 𝐶 ran (Homa𝑐) = ran 𝐻)
7 df-arw 16942 . . . 4 Arrow = (𝑐 ∈ Cat ↦ ran (Homa𝑐))
83fvexi 6389 . . . . . 6 𝐻 ∈ V
98rnex 7298 . . . . 5 ran 𝐻 ∈ V
109uniex 7151 . . . 4 ran 𝐻 ∈ V
116, 7, 10fvmpt 6471 . . 3 (𝐶 ∈ Cat → (Arrow‘𝐶) = ran 𝐻)
127dmmptss 5817 . . . . . . 7 dom Arrow ⊆ Cat
1312sseli 3757 . . . . . 6 (𝐶 ∈ dom Arrow → 𝐶 ∈ Cat)
1413con3i 151 . . . . 5 𝐶 ∈ Cat → ¬ 𝐶 ∈ dom Arrow)
15 ndmfv 6405 . . . . 5 𝐶 ∈ dom Arrow → (Arrow‘𝐶) = ∅)
1614, 15syl 17 . . . 4 𝐶 ∈ Cat → (Arrow‘𝐶) = ∅)
17 df-homa 16941 . . . . . . . . . . . . 13 Homa = (𝑐 ∈ Cat ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥))))
1817dmmptss 5817 . . . . . . . . . . . 12 dom Homa ⊆ Cat
1918sseli 3757 . . . . . . . . . . 11 (𝐶 ∈ dom Homa𝐶 ∈ Cat)
2019con3i 151 . . . . . . . . . 10 𝐶 ∈ Cat → ¬ 𝐶 ∈ dom Homa)
21 ndmfv 6405 . . . . . . . . . 10 𝐶 ∈ dom Homa → (Homa𝐶) = ∅)
2220, 21syl 17 . . . . . . . . 9 𝐶 ∈ Cat → (Homa𝐶) = ∅)
233, 22syl5eq 2811 . . . . . . . 8 𝐶 ∈ Cat → 𝐻 = ∅)
2423rneqd 5521 . . . . . . 7 𝐶 ∈ Cat → ran 𝐻 = ran ∅)
25 rn0 5546 . . . . . . 7 ran ∅ = ∅
2624, 25syl6eq 2815 . . . . . 6 𝐶 ∈ Cat → ran 𝐻 = ∅)
2726unieqd 4604 . . . . 5 𝐶 ∈ Cat → ran 𝐻 = ∅)
28 uni0 4623 . . . . 5 ∅ = ∅
2927, 28syl6eq 2815 . . . 4 𝐶 ∈ Cat → ran 𝐻 = ∅)
3016, 29eqtr4d 2802 . . 3 𝐶 ∈ Cat → (Arrow‘𝐶) = ran 𝐻)
3111, 30pm2.61i 176 . 2 (Arrow‘𝐶) = ran 𝐻
321, 31eqtri 2787 1 𝐴 = ran 𝐻
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1652  wcel 2155  c0 4079  {csn 4334   cuni 4594  cmpt 4888   × cxp 5275  dom cdm 5277  ran crn 5278  cfv 6068  Basecbs 16130  Hom chom 16225  Catccat 16590  Arrowcarw 16937  Homachoma 16938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fv 6076  df-homa 16941  df-arw 16942
This theorem is referenced by:  arwhoma  16960  homarw  16961
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