MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  homadmcd Structured version   Visualization version   GIF version

Theorem homadmcd 17293
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 17287 . . . 4 Rel (𝑋𝐻𝑌)
3 1st2nd 7724 . . . 4 ((Rel (𝑋𝐻𝑌) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → 𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
42, 3mpan 689 . . 3 (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
5 1st2ndbr 7727 . . . . . 6 ((Rel (𝑋𝐻𝑌) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → (1st𝐹)(𝑋𝐻𝑌)(2nd𝐹))
62, 5mpan 689 . . . . 5 (𝐹 ∈ (𝑋𝐻𝑌) → (1st𝐹)(𝑋𝐻𝑌)(2nd𝐹))
71homa1 17288 . . . . 5 ((1st𝐹)(𝑋𝐻𝑌)(2nd𝐹) → (1st𝐹) = ⟨𝑋, 𝑌⟩)
86, 7syl 17 . . . 4 (𝐹 ∈ (𝑋𝐻𝑌) → (1st𝐹) = ⟨𝑋, 𝑌⟩)
98opeq1d 4784 . . 3 (𝐹 ∈ (𝑋𝐻𝑌) → ⟨(1st𝐹), (2nd𝐹)⟩ = ⟨⟨𝑋, 𝑌⟩, (2nd𝐹)⟩)
104, 9eqtrd 2857 . 2 (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 = ⟨⟨𝑋, 𝑌⟩, (2nd𝐹)⟩)
11 df-ot 4548 . 2 𝑋, 𝑌, (2nd𝐹)⟩ = ⟨⟨𝑋, 𝑌⟩, (2nd𝐹)⟩
1210, 11eqtr4di 2875 1 (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 = ⟨𝑋, 𝑌, (2nd𝐹)⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2114  cop 4545  cotp 4547   class class class wbr 5042  Rel wrel 5537  cfv 6334  (class class class)co 7140  1st c1st 7673  2nd c2nd 7674  Homachoma 17274
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-ot 4548  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143  df-1st 7675  df-2nd 7676  df-homa 17277
This theorem is referenced by:  arwdmcd  17303  arwlid  17323  arwrid  17324
  Copyright terms: Public domain W3C validator