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Theorem arwval 16914
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 6353 . . . . . . 7 (𝑐 = 𝐶 → (Homa𝑐) = (Homa𝐶))
3 arwval.h . . . . . . 7 𝐻 = (Homa𝐶)
42, 3syl6eqr 2812 . . . . . 6 (𝑐 = 𝐶 → (Homa𝑐) = 𝐻)
54rneqd 5508 . . . . 5 (𝑐 = 𝐶 → ran (Homa𝑐) = ran 𝐻)
65unieqd 4598 . . . 4 (𝑐 = 𝐶 ran (Homa𝑐) = ran 𝐻)
7 df-arw 16898 . . . 4 Arrow = (𝑐 ∈ Cat ↦ ran (Homa𝑐))
8 fvex 6363 . . . . . . 7 (Homa𝐶) ∈ V
93, 8eqeltri 2835 . . . . . 6 𝐻 ∈ V
109rnex 7266 . . . . 5 ran 𝐻 ∈ V
1110uniex 7119 . . . 4 ran 𝐻 ∈ V
126, 7, 11fvmpt 6445 . . 3 (𝐶 ∈ Cat → (Arrow‘𝐶) = ran 𝐻)
137dmmptss 5792 . . . . . . 7 dom Arrow ⊆ Cat
1413sseli 3740 . . . . . 6 (𝐶 ∈ dom Arrow → 𝐶 ∈ Cat)
1514con3i 150 . . . . 5 𝐶 ∈ Cat → ¬ 𝐶 ∈ dom Arrow)
16 ndmfv 6380 . . . . 5 𝐶 ∈ dom Arrow → (Arrow‘𝐶) = ∅)
1715, 16syl 17 . . . 4 𝐶 ∈ Cat → (Arrow‘𝐶) = ∅)
18 df-homa 16897 . . . . . . . . . . . . 13 Homa = (𝑐 ∈ Cat ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥))))
1918dmmptss 5792 . . . . . . . . . . . 12 dom Homa ⊆ Cat
2019sseli 3740 . . . . . . . . . . 11 (𝐶 ∈ dom Homa𝐶 ∈ Cat)
2120con3i 150 . . . . . . . . . 10 𝐶 ∈ Cat → ¬ 𝐶 ∈ dom Homa)
22 ndmfv 6380 . . . . . . . . . 10 𝐶 ∈ dom Homa → (Homa𝐶) = ∅)
2321, 22syl 17 . . . . . . . . 9 𝐶 ∈ Cat → (Homa𝐶) = ∅)
243, 23syl5eq 2806 . . . . . . . 8 𝐶 ∈ Cat → 𝐻 = ∅)
2524rneqd 5508 . . . . . . 7 𝐶 ∈ Cat → ran 𝐻 = ran ∅)
26 rn0 5532 . . . . . . 7 ran ∅ = ∅
2725, 26syl6eq 2810 . . . . . 6 𝐶 ∈ Cat → ran 𝐻 = ∅)
2827unieqd 4598 . . . . 5 𝐶 ∈ Cat → ran 𝐻 = ∅)
29 uni0 4617 . . . . 5 ∅ = ∅
3028, 29syl6eq 2810 . . . 4 𝐶 ∈ Cat → ran 𝐻 = ∅)
3117, 30eqtr4d 2797 . . 3 𝐶 ∈ Cat → (Arrow‘𝐶) = ran 𝐻)
3212, 31pm2.61i 176 . 2 (Arrow‘𝐶) = ran 𝐻
331, 32eqtri 2782 1 𝐴 = ran 𝐻
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1632  wcel 2139  Vcvv 3340  c0 4058  {csn 4321   cuni 4588  cmpt 4881   × cxp 5264  dom cdm 5266  ran crn 5267  cfv 6049  Basecbs 16079  Hom chom 16174  Catccat 16546  Arrowcarw 16893  Homachoma 16894
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fv 6057  df-homa 16897  df-arw 16898
This theorem is referenced by:  arwhoma  16916  homarw  16917
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