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Mirrors > Home > ILE Home > Th. List > relco | GIF version |
Description: A composition is a relation. Exercise 24 of [TakeutiZaring] p. 25. (Contributed by NM, 26-Jan-1997.) |
Ref | Expression |
---|---|
relco | ⊢ Rel (𝐴 ∘ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-co 4650 | . 2 ⊢ (𝐴 ∘ 𝐵) = {〈𝑥, 𝑦〉 ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)} | |
2 | 1 | relopabi 4767 | 1 ⊢ Rel (𝐴 ∘ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ∃wex 1503 class class class wbr 4018 ∘ ccom 4645 Rel wrel 4646 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-opab 4080 df-xp 4647 df-rel 4648 df-co 4650 |
This theorem is referenced by: dfco2 5143 resco 5148 coiun 5153 cocnvcnv2 5155 cores2 5156 co02 5157 co01 5158 coi1 5159 coass 5162 cossxp 5166 funco 5272 fmptco 5699 cofunexg 6129 dftpos4 6283 |
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