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Theorem homadmcd 18004
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 17998 . . . 4 Rel (𝑋𝐻𝑌)
3 1st2nd 8024 . . . 4 ((Rel (𝑋𝐻𝑌) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → 𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
42, 3mpan 687 . . 3 (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
5 1st2ndbr 8027 . . . . . 6 ((Rel (𝑋𝐻𝑌) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → (1st𝐹)(𝑋𝐻𝑌)(2nd𝐹))
62, 5mpan 687 . . . . 5 (𝐹 ∈ (𝑋𝐻𝑌) → (1st𝐹)(𝑋𝐻𝑌)(2nd𝐹))
71homa1 17999 . . . . 5 ((1st𝐹)(𝑋𝐻𝑌)(2nd𝐹) → (1st𝐹) = ⟨𝑋, 𝑌⟩)
86, 7syl 17 . . . 4 (𝐹 ∈ (𝑋𝐻𝑌) → (1st𝐹) = ⟨𝑋, 𝑌⟩)
98opeq1d 4874 . . 3 (𝐹 ∈ (𝑋𝐻𝑌) → ⟨(1st𝐹), (2nd𝐹)⟩ = ⟨⟨𝑋, 𝑌⟩, (2nd𝐹)⟩)
104, 9eqtrd 2766 . 2 (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 = ⟨⟨𝑋, 𝑌⟩, (2nd𝐹)⟩)
11 df-ot 4632 . 2 𝑋, 𝑌, (2nd𝐹)⟩ = ⟨⟨𝑋, 𝑌⟩, (2nd𝐹)⟩
1210, 11eqtr4di 2784 1 (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 = ⟨𝑋, 𝑌, (2nd𝐹)⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  cop 4629  cotp 4631   class class class wbr 5141  Rel wrel 5674  cfv 6537  (class class class)co 7405  1st c1st 7972  2nd c2nd 7973  Homachoma 17985
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-ot 4632  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-1st 7974  df-2nd 7975  df-homa 17988
This theorem is referenced by:  arwdmcd  18014  arwlid  18034  arwrid  18035
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