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Theorem arwhoma 16902
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 16900 . . . . . 6 𝐴 = ran 𝐻
43eleq2i 2842 . . . . 5 (𝐹𝐴𝐹 ran 𝐻)
54biimpi 206 . . . 4 (𝐹𝐴𝐹 ran 𝐻)
6 eqid 2771 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
71arwrcl 16901 . . . . . 6 (𝐹𝐴𝐶 ∈ Cat)
82, 6, 7homaf 16887 . . . . 5 (𝐹𝐴𝐻:((Base‘𝐶) × (Base‘𝐶))⟶𝒫 (((Base‘𝐶) × (Base‘𝐶)) × V))
9 ffn 6185 . . . . 5 (𝐻:((Base‘𝐶) × (Base‘𝐶))⟶𝒫 (((Base‘𝐶) × (Base‘𝐶)) × V) → 𝐻 Fn ((Base‘𝐶) × (Base‘𝐶)))
10 fnunirn 6654 . . . . 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 6332 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻𝑧) = (𝐻‘⟨𝑥, 𝑦⟩))
14 df-ov 6796 . . . . . 6 (𝑥𝐻𝑦) = (𝐻‘⟨𝑥, 𝑦⟩)
1513, 14syl6eqr 2823 . . . . 5 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻𝑧) = (𝑥𝐻𝑦))
1615eleq2d 2836 . . . 4 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹 ∈ (𝐻𝑧) ↔ 𝐹 ∈ (𝑥𝐻𝑦)))
1716rexxp 5403 . . 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 16897 . . . . . 6 (𝐹 ∈ (𝑥𝐻𝑦) → (doma𝐹) = 𝑥)
212homacd 16898 . . . . . 6 (𝐹 ∈ (𝑥𝐻𝑦) → (coda𝐹) = 𝑦)
2220, 21oveq12d 6811 . . . . 5 (𝐹 ∈ (𝑥𝐻𝑦) → ((doma𝐹)𝐻(coda𝐹)) = (𝑥𝐻𝑦))
2319, 22eleqtrrd 2853 . . . 4 (𝐹 ∈ (𝑥𝐻𝑦) → 𝐹 ∈ ((doma𝐹)𝐻(coda𝐹)))
2423rexlimivw 3177 . . 3 (∃𝑦 ∈ (Base‘𝐶)𝐹 ∈ (𝑥𝐻𝑦) → 𝐹 ∈ ((doma𝐹)𝐻(coda𝐹)))
2524rexlimivw 3177 . 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 1631  wcel 2145  wrex 3062  Vcvv 3351  𝒫 cpw 4297  cop 4322   cuni 4574   × cxp 5247  ran crn 5250   Fn wfn 6026  wf 6027  cfv 6031  (class class class)co 6793  Basecbs 16064  domacdoma 16877  codaccoda 16878  Arrowcarw 16879  Homachoma 16880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-1st 7315  df-2nd 7316  df-doma 16881  df-coda 16882  df-homa 16883  df-arw 16884
This theorem is referenced by:  arwdm  16904  arwcd  16905  arwhom  16908  arwdmcd  16909  coapm  16928
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