Theorem relco 5266

Description: A composition is a relation. Exercise 24 of [TakeutiZaring] p. 25. (Contributed by NM, 26-Jan-1997.)

Assertion

Ref	Expression
relco	⊢ Rel (𝐴 ∘ 𝐵)

Proof of Theorem relco

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-co 4763	. 2 ⊢ (𝐴 ∘ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)}
2	1	relopabi 4885	1 ⊢ Rel (𝐴 ∘ 𝐵)

Syntax hints:   ∧ wa 104  ∃wex 1541   class class class wbr 4114   ∘ ccom 4758  Rel wrel 4759

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-opab 4177  df-xp 4760  df-rel 4761  df-co 4763

This theorem is referenced by:  dfco2  5267  resco  5272  coiun  5277  cocnvcnv2  5279  cores2  5280  co02  5281  co01  5282  coi1  5283  coass  5286  cossxp  5290  funco  5397  fmptco  5848  cofunexg  6311  dftpos4  6507  znleval  14927