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Theorem homacd 18063
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 18047 . . . 4 coda = (2nd ∘ 1st )
21fveq1i 6902 . . 3 (coda𝐹) = ((2nd ∘ 1st )‘𝐹)
3 fo1st 8023 . . . . 5 1st :V–onto→V
4 fof 6815 . . . . 5 (1st :V–onto→V → 1st :V⟶V)
53, 4ax-mp 5 . . . 4 1st :V⟶V
6 elex 3482 . . . 4 (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 ∈ V)
7 fvco3 7001 . . . 4 ((1st :V⟶V ∧ 𝐹 ∈ V) → ((2nd ∘ 1st )‘𝐹) = (2nd ‘(1st𝐹)))
85, 6, 7sylancr 585 . . 3 (𝐹 ∈ (𝑋𝐻𝑌) → ((2nd ∘ 1st )‘𝐹) = (2nd ‘(1st𝐹)))
92, 8eqtrid 2778 . 2 (𝐹 ∈ (𝑋𝐻𝑌) → (coda𝐹) = (2nd ‘(1st𝐹)))
10 homahom.h . . . . . 6 𝐻 = (Homa𝐶)
1110homarel 18058 . . . . 5 Rel (𝑋𝐻𝑌)
12 1st2ndbr 8056 . . . . 5 ((Rel (𝑋𝐻𝑌) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → (1st𝐹)(𝑋𝐻𝑌)(2nd𝐹))
1311, 12mpan 688 . . . 4 (𝐹 ∈ (𝑋𝐻𝑌) → (1st𝐹)(𝑋𝐻𝑌)(2nd𝐹))
1410homa1 18059 . . . 4 ((1st𝐹)(𝑋𝐻𝑌)(2nd𝐹) → (1st𝐹) = ⟨𝑋, 𝑌⟩)
1513, 14syl 17 . . 3 (𝐹 ∈ (𝑋𝐻𝑌) → (1st𝐹) = ⟨𝑋, 𝑌⟩)
1615fveq2d 6905 . 2 (𝐹 ∈ (𝑋𝐻𝑌) → (2nd ‘(1st𝐹)) = (2nd ‘⟨𝑋, 𝑌⟩))
17 eqid 2726 . . . 4 (Base‘𝐶) = (Base‘𝐶)
1810, 17homarcl2 18057 . . 3 (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
19 op2ndg 8016 . . 3 ((𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
2018, 19syl 17 . 2 (𝐹 ∈ (𝑋𝐻𝑌) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
219, 16, 203eqtrd 2770 1 (𝐹 ∈ (𝑋𝐻𝑌) → (coda𝐹) = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  wcel 2099  Vcvv 3462  cop 4639   class class class wbr 5153  ccom 5686  Rel wrel 5687  wf 6550  ontowfo 6552  cfv 6554  (class class class)co 7424  1st c1st 8001  2nd c2nd 8002  Basecbs 17213  codaccoda 18043  Homachoma 18045
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-ov 7427  df-1st 8003  df-2nd 8004  df-coda 18047  df-homa 18048
This theorem is referenced by:  arwhoma  18067  idacd  18084  homdmcoa  18089  coaval  18090
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