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Theorem homacd 16460
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 16444 . . . 4 coda = (2nd ∘ 1st )
21fveq1i 6089 . . 3 (coda𝐹) = ((2nd ∘ 1st )‘𝐹)
3 fo1st 7056 . . . . 5 1st :V–onto→V
4 fof 6013 . . . . 5 (1st :V–onto→V → 1st :V⟶V)
53, 4ax-mp 5 . . . 4 1st :V⟶V
6 elex 3184 . . . 4 (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 ∈ V)
7 fvco3 6170 . . . 4 ((1st :V⟶V ∧ 𝐹 ∈ V) → ((2nd ∘ 1st )‘𝐹) = (2nd ‘(1st𝐹)))
85, 6, 7sylancr 693 . . 3 (𝐹 ∈ (𝑋𝐻𝑌) → ((2nd ∘ 1st )‘𝐹) = (2nd ‘(1st𝐹)))
92, 8syl5eq 2655 . 2 (𝐹 ∈ (𝑋𝐻𝑌) → (coda𝐹) = (2nd ‘(1st𝐹)))
10 homahom.h . . . . . 6 𝐻 = (Homa𝐶)
1110homarel 16455 . . . . 5 Rel (𝑋𝐻𝑌)
12 1st2ndbr 7085 . . . . 5 ((Rel (𝑋𝐻𝑌) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → (1st𝐹)(𝑋𝐻𝑌)(2nd𝐹))
1311, 12mpan 701 . . . 4 (𝐹 ∈ (𝑋𝐻𝑌) → (1st𝐹)(𝑋𝐻𝑌)(2nd𝐹))
1410homa1 16456 . . . 4 ((1st𝐹)(𝑋𝐻𝑌)(2nd𝐹) → (1st𝐹) = ⟨𝑋, 𝑌⟩)
1513, 14syl 17 . . 3 (𝐹 ∈ (𝑋𝐻𝑌) → (1st𝐹) = ⟨𝑋, 𝑌⟩)
1615fveq2d 6092 . 2 (𝐹 ∈ (𝑋𝐻𝑌) → (2nd ‘(1st𝐹)) = (2nd ‘⟨𝑋, 𝑌⟩))
17 eqid 2609 . . . 4 (Base‘𝐶) = (Base‘𝐶)
1810, 17homarcl2 16454 . . 3 (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
19 op2ndg 7049 . . 3 ((𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
2018, 19syl 17 . 2 (𝐹 ∈ (𝑋𝐻𝑌) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
219, 16, 203eqtrd 2647 1 (𝐹 ∈ (𝑋𝐻𝑌) → (coda𝐹) = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  Vcvv 3172  cop 4130   class class class wbr 4577  ccom 5032  Rel wrel 5033  wf 5786  ontowfo 5788  cfv 5790  (class class class)co 6527  1st c1st 7034  2nd c2nd 7035  Basecbs 15641  codaccoda 16440  Homachoma 16442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-1st 7036  df-2nd 7037  df-coda 16444  df-homa 16445
This theorem is referenced by:  arwhoma  16464  idacd  16481  homdmcoa  16486  coaval  16487
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