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Theorem homacd 18086
Description: The codomain 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
homacd (𝐹 ∈ (𝑋𝐻𝑌) → (coda𝐹) = 𝑌)

Proof of Theorem homacd
StepHypRef Expression
1 df-coda 18070 . . . 4 coda = (2nd ∘ 1st )
21fveq1i 6907 . . 3 (coda𝐹) = ((2nd ∘ 1st )‘𝐹)
3 fo1st 8034 . . . . 5 1st :V–onto→V
4 fof 6820 . . . . 5 (1st :V–onto→V → 1st :V⟶V)
53, 4ax-mp 5 . . . 4 1st :V⟶V
6 elex 3501 . . . 4 (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 ∈ V)
7 fvco3 7008 . . . 4 ((1st :V⟶V ∧ 𝐹 ∈ V) → ((2nd ∘ 1st )‘𝐹) = (2nd ‘(1st𝐹)))
85, 6, 7sylancr 587 . . 3 (𝐹 ∈ (𝑋𝐻𝑌) → ((2nd ∘ 1st )‘𝐹) = (2nd ‘(1st𝐹)))
92, 8eqtrid 2789 . 2 (𝐹 ∈ (𝑋𝐻𝑌) → (coda𝐹) = (2nd ‘(1st𝐹)))
10 homahom.h . . . . . 6 𝐻 = (Homa𝐶)
1110homarel 18081 . . . . 5 Rel (𝑋𝐻𝑌)
12 1st2ndbr 8067 . . . . 5 ((Rel (𝑋𝐻𝑌) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → (1st𝐹)(𝑋𝐻𝑌)(2nd𝐹))
1311, 12mpan 690 . . . 4 (𝐹 ∈ (𝑋𝐻𝑌) → (1st𝐹)(𝑋𝐻𝑌)(2nd𝐹))
1410homa1 18082 . . . 4 ((1st𝐹)(𝑋𝐻𝑌)(2nd𝐹) → (1st𝐹) = ⟨𝑋, 𝑌⟩)
1513, 14syl 17 . . 3 (𝐹 ∈ (𝑋𝐻𝑌) → (1st𝐹) = ⟨𝑋, 𝑌⟩)
1615fveq2d 6910 . 2 (𝐹 ∈ (𝑋𝐻𝑌) → (2nd ‘(1st𝐹)) = (2nd ‘⟨𝑋, 𝑌⟩))
17 eqid 2737 . . . 4 (Base‘𝐶) = (Base‘𝐶)
1810, 17homarcl2 18080 . . 3 (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
19 op2ndg 8027 . . 3 ((𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
2018, 19syl 17 . 2 (𝐹 ∈ (𝑋𝐻𝑌) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
219, 16, 203eqtrd 2781 1 (𝐹 ∈ (𝑋𝐻𝑌) → (coda𝐹) = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  cop 4632   class class class wbr 5143  ccom 5689  Rel wrel 5690  wf 6557  ontowfo 6559  cfv 6561  (class class class)co 7431  1st c1st 8012  2nd c2nd 8013  Basecbs 17247  codaccoda 18066  Homachoma 18068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-1st 8014  df-2nd 8015  df-coda 18070  df-homa 18071
This theorem is referenced by:  arwhoma  18090  idacd  18107  homdmcoa  18112  coaval  18113
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