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Theorem arwhoma 16916
Description: An arrow is contained in the hom-set corresponding to its domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
arwrcl.a 𝐴 = (Arrow‘𝐶)
arwhoma.h 𝐻 = (Homa𝐶)
Assertion
Ref Expression
arwhoma (𝐹𝐴𝐹 ∈ ((doma𝐹)𝐻(coda𝐹)))

Proof of Theorem arwhoma
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 arwrcl.a . . . . . . 7 𝐴 = (Arrow‘𝐶)
2 arwhoma.h . . . . . . 7 𝐻 = (Homa𝐶)
31, 2arwval 16914 . . . . . 6 𝐴 = ran 𝐻
43eleq2i 2831 . . . . 5 (𝐹𝐴𝐹 ran 𝐻)
54biimpi 206 . . . 4 (𝐹𝐴𝐹 ran 𝐻)
6 eqid 2760 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
71arwrcl 16915 . . . . . 6 (𝐹𝐴𝐶 ∈ Cat)
82, 6, 7homaf 16901 . . . . 5 (𝐹𝐴𝐻:((Base‘𝐶) × (Base‘𝐶))⟶𝒫 (((Base‘𝐶) × (Base‘𝐶)) × V))
9 ffn 6206 . . . . 5 (𝐻:((Base‘𝐶) × (Base‘𝐶))⟶𝒫 (((Base‘𝐶) × (Base‘𝐶)) × V) → 𝐻 Fn ((Base‘𝐶) × (Base‘𝐶)))
10 fnunirn 6675 . . . . 5 (𝐻 Fn ((Base‘𝐶) × (Base‘𝐶)) → (𝐹 ran 𝐻 ↔ ∃𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))𝐹 ∈ (𝐻𝑧)))
118, 9, 103syl 18 . . . 4 (𝐹𝐴 → (𝐹 ran 𝐻 ↔ ∃𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))𝐹 ∈ (𝐻𝑧)))
125, 11mpbid 222 . . 3 (𝐹𝐴 → ∃𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))𝐹 ∈ (𝐻𝑧))
13 fveq2 6353 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻𝑧) = (𝐻‘⟨𝑥, 𝑦⟩))
14 df-ov 6817 . . . . . 6 (𝑥𝐻𝑦) = (𝐻‘⟨𝑥, 𝑦⟩)
1513, 14syl6eqr 2812 . . . . 5 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻𝑧) = (𝑥𝐻𝑦))
1615eleq2d 2825 . . . 4 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹 ∈ (𝐻𝑧) ↔ 𝐹 ∈ (𝑥𝐻𝑦)))
1716rexxp 5420 . . 3 (∃𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))𝐹 ∈ (𝐻𝑧) ↔ ∃𝑥 ∈ (Base‘𝐶)∃𝑦 ∈ (Base‘𝐶)𝐹 ∈ (𝑥𝐻𝑦))
1812, 17sylib 208 . 2 (𝐹𝐴 → ∃𝑥 ∈ (Base‘𝐶)∃𝑦 ∈ (Base‘𝐶)𝐹 ∈ (𝑥𝐻𝑦))
19 id 22 . . . . 5 (𝐹 ∈ (𝑥𝐻𝑦) → 𝐹 ∈ (𝑥𝐻𝑦))
202homadm 16911 . . . . . 6 (𝐹 ∈ (𝑥𝐻𝑦) → (doma𝐹) = 𝑥)
212homacd 16912 . . . . . 6 (𝐹 ∈ (𝑥𝐻𝑦) → (coda𝐹) = 𝑦)
2220, 21oveq12d 6832 . . . . 5 (𝐹 ∈ (𝑥𝐻𝑦) → ((doma𝐹)𝐻(coda𝐹)) = (𝑥𝐻𝑦))
2319, 22eleqtrrd 2842 . . . 4 (𝐹 ∈ (𝑥𝐻𝑦) → 𝐹 ∈ ((doma𝐹)𝐻(coda𝐹)))
2423rexlimivw 3167 . . 3 (∃𝑦 ∈ (Base‘𝐶)𝐹 ∈ (𝑥𝐻𝑦) → 𝐹 ∈ ((doma𝐹)𝐻(coda𝐹)))
2524rexlimivw 3167 . 2 (∃𝑥 ∈ (Base‘𝐶)∃𝑦 ∈ (Base‘𝐶)𝐹 ∈ (𝑥𝐻𝑦) → 𝐹 ∈ ((doma𝐹)𝐻(coda𝐹)))
2618, 25syl 17 1 (𝐹𝐴𝐹 ∈ ((doma𝐹)𝐻(coda𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1632  wcel 2139  wrex 3051  Vcvv 3340  𝒫 cpw 4302  cop 4327   cuni 4588   × cxp 5264  ran crn 5267   Fn wfn 6044  wf 6045  cfv 6049  (class class class)co 6814  Basecbs 16079  domacdoma 16891  codaccoda 16892  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-rep 4923  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-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  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-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-1st 7334  df-2nd 7335  df-doma 16895  df-coda 16896  df-homa 16897  df-arw 16898
This theorem is referenced by:  arwdm  16918  arwcd  16919  arwhom  16922  arwdmcd  16923  coapm  16942
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