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Theorem homadm 16911
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 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  ontowfo 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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