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Theorem homadm 16462
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)

Proof of Theorem homadm
StepHypRef Expression
1 df-doma 16446 . . . 4 doma = (1st ∘ 1st )
21fveq1i 6089 . . 3 (doma𝐹) = ((1st ∘ 1st )‘𝐹)
3 fo1st 7057 . . . . 5 1st :V–onto→V
4 fof 6013 . . . . 5 (1st :V–onto→V → 1st :V⟶V)
53, 4ax-mp 5 . . . 4 1st :V⟶V
6 elex 3185 . . . 4 (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 ∈ V)
7 fvco3 6170 . . . 4 ((1st :V⟶V ∧ 𝐹 ∈ V) → ((1st ∘ 1st )‘𝐹) = (1st ‘(1st𝐹)))
85, 6, 7sylancr 694 . . 3 (𝐹 ∈ (𝑋𝐻𝑌) → ((1st ∘ 1st )‘𝐹) = (1st ‘(1st𝐹)))
92, 8syl5eq 2656 . 2 (𝐹 ∈ (𝑋𝐻𝑌) → (doma𝐹) = (1st ‘(1st𝐹)))
10 homahom.h . . . . . 6 𝐻 = (Homa𝐶)
1110homarel 16458 . . . . 5 Rel (𝑋𝐻𝑌)
12 1st2ndbr 7086 . . . . 5 ((Rel (𝑋𝐻𝑌) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → (1st𝐹)(𝑋𝐻𝑌)(2nd𝐹))
1311, 12mpan 702 . . . 4 (𝐹 ∈ (𝑋𝐻𝑌) → (1st𝐹)(𝑋𝐻𝑌)(2nd𝐹))
1410homa1 16459 . . . 4 ((1st𝐹)(𝑋𝐻𝑌)(2nd𝐹) → (1st𝐹) = ⟨𝑋, 𝑌⟩)
1513, 14syl 17 . . 3 (𝐹 ∈ (𝑋𝐻𝑌) → (1st𝐹) = ⟨𝑋, 𝑌⟩)
1615fveq2d 6092 . 2 (𝐹 ∈ (𝑋𝐻𝑌) → (1st ‘(1st𝐹)) = (1st ‘⟨𝑋, 𝑌⟩))
17 eqid 2610 . . . 4 (Base‘𝐶) = (Base‘𝐶)
1810, 17homarcl2 16457 . . 3 (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
19 op1stg 7049 . . 3 ((𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
2018, 19syl 17 . 2 (𝐹 ∈ (𝑋𝐻𝑌) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
219, 16, 203eqtrd 2648 1 (𝐹 ∈ (𝑋𝐻𝑌) → (doma𝐹) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  Vcvv 3173  cop 4131   class class class wbr 4578  ccom 5032  Rel wrel 5033  wf 5786  ontowfo 5788  cfv 5790  (class class class)co 6527  1st c1st 7035  2nd c2nd 7036  Basecbs 15644  domacdoma 16442  Homachoma 16445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4694  ax-sep 4704  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4368  df-iun 4452  df-br 4579  df-opab 4639  df-mpt 4640  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-1st 7037  df-2nd 7038  df-doma 16446  df-homa 16448
This theorem is referenced by:  arwhoma  16467  idadm  16483  homdmcoa  16489  coaval  16490
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