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Theorem homadmcd 18028
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 18022 . . . 4 Rel (𝑋𝐻𝑌)
3 1st2nd 8039 . . . 4 ((Rel (𝑋𝐻𝑌) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → 𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
42, 3mpan 688 . . 3 (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
5 1st2ndbr 8042 . . . . . 6 ((Rel (𝑋𝐻𝑌) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → (1st𝐹)(𝑋𝐻𝑌)(2nd𝐹))
62, 5mpan 688 . . . . 5 (𝐹 ∈ (𝑋𝐻𝑌) → (1st𝐹)(𝑋𝐻𝑌)(2nd𝐹))
71homa1 18023 . . . . 5 ((1st𝐹)(𝑋𝐻𝑌)(2nd𝐹) → (1st𝐹) = ⟨𝑋, 𝑌⟩)
86, 7syl 17 . . . 4 (𝐹 ∈ (𝑋𝐻𝑌) → (1st𝐹) = ⟨𝑋, 𝑌⟩)
98opeq1d 4873 . . 3 (𝐹 ∈ (𝑋𝐻𝑌) → ⟨(1st𝐹), (2nd𝐹)⟩ = ⟨⟨𝑋, 𝑌⟩, (2nd𝐹)⟩)
104, 9eqtrd 2765 . 2 (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 = ⟨⟨𝑋, 𝑌⟩, (2nd𝐹)⟩)
11 df-ot 4631 . 2 𝑋, 𝑌, (2nd𝐹)⟩ = ⟨⟨𝑋, 𝑌⟩, (2nd𝐹)⟩
1210, 11eqtr4di 2783 1 (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 = ⟨𝑋, 𝑌, (2nd𝐹)⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  cop 4628  cotp 4630   class class class wbr 5141  Rel wrel 5675  cfv 6541  (class class class)co 7414  1st c1st 7987  2nd c2nd 7988  Homachoma 18009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5357  ax-pr 5421  ax-un 7736
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4317  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-ot 4631  df-uni 4902  df-iun 4991  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5568  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ov 7417  df-1st 7989  df-2nd 7990  df-homa 18012
This theorem is referenced by:  arwdmcd  18038  arwlid  18058  arwrid  18059
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