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 Description: The domain of an arrow with known domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypothesis
Ref Expression
homahom.h 𝐻 = (Homa𝐶)
Assertion
Ref Expression
homadm (𝐹 ∈ (𝑋𝐻𝑌) → (doma𝐹) = 𝑋)

StepHypRef Expression
1 df-doma 16895 . . . 4 doma = (1st ∘ 1st )
21fveq1i 6354 . . 3 (doma𝐹) = ((1st ∘ 1st )‘𝐹)
3 fo1st 7354 . . . . 5 1st :V–onto→V
4 fof 6277 . . . . 5 (1st :V–onto→V → 1st :V⟶V)
53, 4ax-mp 5 . . . 4 1st :V⟶V
6 elex 3352 . . . 4 (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 ∈ V)
7 fvco3 6438 . . . 4 ((1st :V⟶V ∧ 𝐹 ∈ V) → ((1st ∘ 1st )‘𝐹) = (1st ‘(1st𝐹)))
85, 6, 7sylancr 698 . . 3 (𝐹 ∈ (𝑋𝐻𝑌) → ((1st ∘ 1st )‘𝐹) = (1st ‘(1st𝐹)))
92, 8syl5eq 2806 . 2 (𝐹 ∈ (𝑋𝐻𝑌) → (doma𝐹) = (1st ‘(1st𝐹)))
10 homahom.h . . . . . 6 𝐻 = (Homa𝐶)
1110homarel 16907 . . . . 5 Rel (𝑋𝐻𝑌)
12 1st2ndbr 7385 . . . . 5 ((Rel (𝑋𝐻𝑌) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → (1st𝐹)(𝑋𝐻𝑌)(2nd𝐹))
1311, 12mpan 708 . . . 4 (𝐹 ∈ (𝑋𝐻𝑌) → (1st𝐹)(𝑋𝐻𝑌)(2nd𝐹))
1410homa1 16908 . . . 4 ((1st𝐹)(𝑋𝐻𝑌)(2nd𝐹) → (1st𝐹) = ⟨𝑋, 𝑌⟩)
1513, 14syl 17 . . 3 (𝐹 ∈ (𝑋𝐻𝑌) → (1st𝐹) = ⟨𝑋, 𝑌⟩)
1615fveq2d 6357 . 2 (𝐹 ∈ (𝑋𝐻𝑌) → (1st ‘(1st𝐹)) = (1st ‘⟨𝑋, 𝑌⟩))
17 eqid 2760 . . . 4 (Base‘𝐶) = (Base‘𝐶)
1810, 17homarcl2 16906 . . 3 (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
19 op1stg 7346 . . 3 ((𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
2018, 19syl 17 . 2 (𝐹 ∈ (𝑋𝐻𝑌) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
219, 16, 203eqtrd 2798 1 (𝐹 ∈ (𝑋𝐻𝑌) → (doma𝐹) = 𝑋)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1632   ∈ wcel 2139  Vcvv 3340  ⟨cop 4327   class class class wbr 4804   ∘ ccom 5270  Rel wrel 5271  ⟶wf 6045  –onto→wfo 6047  ‘cfv 6049  (class class class)co 6814  1st c1st 7332  2nd c2nd 7333  Basecbs 16079  domacdoma 16891  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-homa 16897 This theorem is referenced by:  arwhoma  16916  idadm  16932  homdmcoa  16938  coaval  16939
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