Theorem relco 5190

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 4692	. 2 ⊢ (𝐴 ∘ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)}
2	1	relopabi 4811	1 ⊢ Rel (𝐴 ∘ 𝐵)

Syntax hints:   ∧ wa 104  ∃wex 1516   class class class wbr 4051   ∘ ccom 4687  Rel wrel 4688

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-opab 4114  df-xp 4689  df-rel 4690  df-co 4692

This theorem is referenced by:  dfco2  5191  resco  5196  coiun  5201  cocnvcnv2  5203  cores2  5204  co02  5205  co01  5206  coi1  5207  coass  5210  cossxp  5214  funco  5320  fmptco  5759  cofunexg  6207  dftpos4  6362  znleval  14490