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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 17363 . . . 4 doma = (1st ∘ 1st )
21fveq1i 6664 . . 3 (doma𝐹) = ((1st ∘ 1st )‘𝐹)
3 fo1st 7719 . . . . 5 1st :V–onto→V
4 fof 6581 . . . . 5 (1st :V–onto→V → 1st :V⟶V)
53, 4ax-mp 5 . . . 4 1st :V⟶V
6 elex 3428 . . . 4 (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 ∈ V)
7 fvco3 6756 . . . 4 ((1st :V⟶V ∧ 𝐹 ∈ V) → ((1st ∘ 1st )‘𝐹) = (1st ‘(1st𝐹)))
85, 6, 7sylancr 590 . . 3 (𝐹 ∈ (𝑋𝐻𝑌) → ((1st ∘ 1st )‘𝐹) = (1st ‘(1st𝐹)))
92, 8syl5eq 2805 . 2 (𝐹 ∈ (𝑋𝐻𝑌) → (doma𝐹) = (1st ‘(1st𝐹)))
10 homahom.h . . . . . 6 𝐻 = (Homa𝐶)
1110homarel 17375 . . . . 5 Rel (𝑋𝐻𝑌)
12 1st2ndbr 7751 . . . . 5 ((Rel (𝑋𝐻𝑌) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → (1st𝐹)(𝑋𝐻𝑌)(2nd𝐹))
1311, 12mpan 689 . . . 4 (𝐹 ∈ (𝑋𝐻𝑌) → (1st𝐹)(𝑋𝐻𝑌)(2nd𝐹))
1410homa1 17376 . . . 4 ((1st𝐹)(𝑋𝐻𝑌)(2nd𝐹) → (1st𝐹) = ⟨𝑋, 𝑌⟩)
1513, 14syl 17 . . 3 (𝐹 ∈ (𝑋𝐻𝑌) → (1st𝐹) = ⟨𝑋, 𝑌⟩)
1615fveq2d 6667 . 2 (𝐹 ∈ (𝑋𝐻𝑌) → (1st ‘(1st𝐹)) = (1st ‘⟨𝑋, 𝑌⟩))
17 eqid 2758 . . . 4 (Base‘𝐶) = (Base‘𝐶)
1810, 17homarcl2 17374 . . 3 (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
19 op1stg 7711 . . 3 ((𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
2018, 19syl 17 . 2 (𝐹 ∈ (𝑋𝐻𝑌) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
219, 16, 203eqtrd 2797 1 (𝐹 ∈ (𝑋𝐻𝑌) → (doma𝐹) = 𝑋)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3409  ⟨cop 4531   class class class wbr 5036   ∘ ccom 5532  Rel wrel 5533  ⟶wf 6336  –onto→wfo 6338  ‘cfv 6340  (class class class)co 7156  1st c1st 7697  2nd c2nd 7698  Basecbs 16554  domacdoma 17359  Homachoma 17362 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7159  df-1st 7699  df-2nd 7700  df-doma 17363  df-homa 17365 This theorem is referenced by:  arwhoma  17384  idadm  17400  homdmcoa  17406  coaval  17407
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