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Theorem arwdmcd 16471
Description: Decompose an arrow into domain, codomain, and morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypothesis
Ref Expression
arwrcl.a 𝐴 = (Arrow‘𝐶)
Assertion
Ref Expression
arwdmcd (𝐹𝐴𝐹 = ⟨(doma𝐹), (coda𝐹), (2nd𝐹)⟩)

Proof of Theorem arwdmcd
StepHypRef Expression
1 arwrcl.a . . 3 𝐴 = (Arrow‘𝐶)
2 eqid 2609 . . 3 (Homa𝐶) = (Homa𝐶)
31, 2arwhoma 16464 . 2 (𝐹𝐴𝐹 ∈ ((doma𝐹)(Homa𝐶)(coda𝐹)))
42homadmcd 16461 . 2 (𝐹 ∈ ((doma𝐹)(Homa𝐶)(coda𝐹)) → 𝐹 = ⟨(doma𝐹), (coda𝐹), (2nd𝐹)⟩)
53, 4syl 17 1 (𝐹𝐴𝐹 = ⟨(doma𝐹), (coda𝐹), (2nd𝐹)⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1976  cotp 4132  cfv 5790  (class class class)co 6527  2nd c2nd 7035  domacdoma 16439  codaccoda 16440  Arrowcarw 16441  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-doma 16443  df-coda 16444  df-homa 16445  df-arw 16446
This theorem is referenced by: (None)
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