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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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