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Theorem homacd 16898
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 16882 . . . 4 coda = (2nd ∘ 1st )
21fveq1i 6333 . . 3 (coda𝐹) = ((2nd ∘ 1st )‘𝐹)
3 fo1st 7335 . . . . 5 1st :V–onto→V
4 fof 6256 . . . . 5 (1st :V–onto→V → 1st :V⟶V)
53, 4ax-mp 5 . . . 4 1st :V⟶V
6 elex 3364 . . . 4 (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 ∈ V)
7 fvco3 6417 . . . 4 ((1st :V⟶V ∧ 𝐹 ∈ V) → ((2nd ∘ 1st )‘𝐹) = (2nd ‘(1st𝐹)))
85, 6, 7sylancr 575 . . 3 (𝐹 ∈ (𝑋𝐻𝑌) → ((2nd ∘ 1st )‘𝐹) = (2nd ‘(1st𝐹)))
92, 8syl5eq 2817 . 2 (𝐹 ∈ (𝑋𝐻𝑌) → (coda𝐹) = (2nd ‘(1st𝐹)))
10 homahom.h . . . . . 6 𝐻 = (Homa𝐶)
1110homarel 16893 . . . . 5 Rel (𝑋𝐻𝑌)
12 1st2ndbr 7366 . . . . 5 ((Rel (𝑋𝐻𝑌) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → (1st𝐹)(𝑋𝐻𝑌)(2nd𝐹))
1311, 12mpan 670 . . . 4 (𝐹 ∈ (𝑋𝐻𝑌) → (1st𝐹)(𝑋𝐻𝑌)(2nd𝐹))
1410homa1 16894 . . . 4 ((1st𝐹)(𝑋𝐻𝑌)(2nd𝐹) → (1st𝐹) = ⟨𝑋, 𝑌⟩)
1513, 14syl 17 . . 3 (𝐹 ∈ (𝑋𝐻𝑌) → (1st𝐹) = ⟨𝑋, 𝑌⟩)
1615fveq2d 6336 . 2 (𝐹 ∈ (𝑋𝐻𝑌) → (2nd ‘(1st𝐹)) = (2nd ‘⟨𝑋, 𝑌⟩))
17 eqid 2771 . . . 4 (Base‘𝐶) = (Base‘𝐶)
1810, 17homarcl2 16892 . . 3 (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
19 op2ndg 7328 . . 3 ((𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
2018, 19syl 17 . 2 (𝐹 ∈ (𝑋𝐻𝑌) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
219, 16, 203eqtrd 2809 1 (𝐹 ∈ (𝑋𝐻𝑌) → (coda𝐹) = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  Vcvv 3351  cop 4322   class class class wbr 4786  ccom 5253  Rel wrel 5254  wf 6027  ontowfo 6029  cfv 6031  (class class class)co 6793  1st c1st 7313  2nd c2nd 7314  Basecbs 16064  codaccoda 16878  Homachoma 16880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-1st 7315  df-2nd 7316  df-coda 16882  df-homa 16883
This theorem is referenced by:  arwhoma  16902  idacd  16919  homdmcoa  16924  coaval  16925
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