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Theorem homacd 16738
 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 16722 . . . 4 coda = (2nd ∘ 1st )
21fveq1i 6230 . . 3 (coda𝐹) = ((2nd ∘ 1st )‘𝐹)
3 fo1st 7230 . . . . 5 1st :V–onto→V
4 fof 6153 . . . . 5 (1st :V–onto→V → 1st :V⟶V)
53, 4ax-mp 5 . . . 4 1st :V⟶V
6 elex 3243 . . . 4 (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 ∈ V)
7 fvco3 6314 . . . 4 ((1st :V⟶V ∧ 𝐹 ∈ V) → ((2nd ∘ 1st )‘𝐹) = (2nd ‘(1st𝐹)))
85, 6, 7sylancr 696 . . 3 (𝐹 ∈ (𝑋𝐻𝑌) → ((2nd ∘ 1st )‘𝐹) = (2nd ‘(1st𝐹)))
92, 8syl5eq 2697 . 2 (𝐹 ∈ (𝑋𝐻𝑌) → (coda𝐹) = (2nd ‘(1st𝐹)))
10 homahom.h . . . . . 6 𝐻 = (Homa𝐶)
1110homarel 16733 . . . . 5 Rel (𝑋𝐻𝑌)
12 1st2ndbr 7261 . . . . 5 ((Rel (𝑋𝐻𝑌) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → (1st𝐹)(𝑋𝐻𝑌)(2nd𝐹))
1311, 12mpan 706 . . . 4 (𝐹 ∈ (𝑋𝐻𝑌) → (1st𝐹)(𝑋𝐻𝑌)(2nd𝐹))
1410homa1 16734 . . . 4 ((1st𝐹)(𝑋𝐻𝑌)(2nd𝐹) → (1st𝐹) = ⟨𝑋, 𝑌⟩)
1513, 14syl 17 . . 3 (𝐹 ∈ (𝑋𝐻𝑌) → (1st𝐹) = ⟨𝑋, 𝑌⟩)
1615fveq2d 6233 . 2 (𝐹 ∈ (𝑋𝐻𝑌) → (2nd ‘(1st𝐹)) = (2nd ‘⟨𝑋, 𝑌⟩))
17 eqid 2651 . . . 4 (Base‘𝐶) = (Base‘𝐶)
1810, 17homarcl2 16732 . . 3 (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
19 op2ndg 7223 . . 3 ((𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
2018, 19syl 17 . 2 (𝐹 ∈ (𝑋𝐻𝑌) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
219, 16, 203eqtrd 2689 1 (𝐹 ∈ (𝑋𝐻𝑌) → (coda𝐹) = 𝑌)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1523   ∈ wcel 2030  Vcvv 3231  ⟨cop 4216   class class class wbr 4685   ∘ ccom 5147  Rel wrel 5148  ⟶wf 5922  –onto→wfo 5924  ‘cfv 5926  (class class class)co 6690  1st c1st 7208  2nd c2nd 7209  Basecbs 15904  codaccoda 16718  Homachoma 16720 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-1st 7210  df-2nd 7211  df-coda 16722  df-homa 16723 This theorem is referenced by:  arwhoma  16742  idacd  16759  homdmcoa  16764  coaval  16765
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