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Theorem arwdmcd 16902
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 2806 . . 3 (Homa𝐶) = (Homa𝐶)
31, 2arwhoma 16895 . 2 (𝐹𝐴𝐹 ∈ ((doma𝐹)(Homa𝐶)(coda𝐹)))
42homadmcd 16892 . 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 1637  wcel 2156  cotp 4378  cfv 6097  (class class class)co 6870  2nd c2nd 7393  domacdoma 16870  codaccoda 16871  Arrowcarw 16872  Homachoma 16873
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7175
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-ot 4379  df-uni 4631  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6060  df-fun 6099  df-fn 6100  df-f 6101  df-f1 6102  df-fo 6103  df-f1o 6104  df-fv 6105  df-ov 6873  df-1st 7394  df-2nd 7395  df-doma 16874  df-coda 16875  df-homa 16876  df-arw 16877
This theorem is referenced by: (None)
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