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Theorem homadmcd 16461
Description: Decompose an arrow into domain, codomain, and morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypothesis
Ref Expression
homahom.h 𝐻 = (Homa𝐶)
Assertion
Ref Expression
homadmcd (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 = ⟨𝑋, 𝑌, (2nd𝐹)⟩)

Proof of Theorem homadmcd
StepHypRef Expression
1 homahom.h . . . . 5 𝐻 = (Homa𝐶)
21homarel 16455 . . . 4 Rel (𝑋𝐻𝑌)
3 1st2nd 7082 . . . 4 ((Rel (𝑋𝐻𝑌) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → 𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
42, 3mpan 701 . . 3 (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
5 1st2ndbr 7085 . . . . . 6 ((Rel (𝑋𝐻𝑌) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → (1st𝐹)(𝑋𝐻𝑌)(2nd𝐹))
62, 5mpan 701 . . . . 5 (𝐹 ∈ (𝑋𝐻𝑌) → (1st𝐹)(𝑋𝐻𝑌)(2nd𝐹))
71homa1 16456 . . . . 5 ((1st𝐹)(𝑋𝐻𝑌)(2nd𝐹) → (1st𝐹) = ⟨𝑋, 𝑌⟩)
86, 7syl 17 . . . 4 (𝐹 ∈ (𝑋𝐻𝑌) → (1st𝐹) = ⟨𝑋, 𝑌⟩)
98opeq1d 4340 . . 3 (𝐹 ∈ (𝑋𝐻𝑌) → ⟨(1st𝐹), (2nd𝐹)⟩ = ⟨⟨𝑋, 𝑌⟩, (2nd𝐹)⟩)
104, 9eqtrd 2643 . 2 (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 = ⟨⟨𝑋, 𝑌⟩, (2nd𝐹)⟩)
11 df-ot 4133 . 2 𝑋, 𝑌, (2nd𝐹)⟩ = ⟨⟨𝑋, 𝑌⟩, (2nd𝐹)⟩
1210, 11syl6eqr 2661 1 (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 = ⟨𝑋, 𝑌, (2nd𝐹)⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1976  cop 4130  cotp 4132   class class class wbr 4577  Rel wrel 5033  cfv 5790  (class class class)co 6527  1st c1st 7034  2nd c2nd 7035  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-ot 4133  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-homa 16445
This theorem is referenced by:  arwdmcd  16471  arwlid  16491  arwrid  16492
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