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Theorem relco 4846
 Description: A composition is a relation. Exercise 24 of [TakeutiZaring] p. 25. (Contributed by NM, 26-Jan-1997.)
Assertion
Ref Expression
relco Rel (𝐴𝐵)

Proof of Theorem relco
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-co 4381 . 2 (𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦)}
21relopabi 4490 1 Rel (𝐴𝐵)
 Colors of variables: wff set class Syntax hints:   ∧ wa 101  ∃wex 1397   class class class wbr 3791   ∘ ccom 4376  Rel wrel 4377 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-opab 3846  df-xp 4378  df-rel 4379  df-co 4381 This theorem is referenced by:  dfco2  4847  resco  4852  coiun  4857  cocnvcnv2  4859  cores2  4860  co02  4861  co01  4862  coi1  4863  coass  4866  cossxp  4870  funco  4967  fmptco  5357  cofunexg  5765  dftpos4  5908
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