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Mirrors > Home > MPE Home > Th. List > relco | Structured version Visualization version 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 5557 | . 2 ⊢ (𝐴 ∘ 𝐵) = {〈𝑥, 𝑦〉 ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)} | |
2 | 1 | relopabi 5687 | 1 ⊢ Rel (𝐴 ∘ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 ∃wex 1771 class class class wbr 5057 ∘ ccom 5552 Rel wrel 5553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-opab 5120 df-xp 5554 df-rel 5555 df-co 5557 |
This theorem is referenced by: dfco2 6091 resco 6096 coeq0 6101 coiun 6102 cocnvcnv2 6104 cores2 6105 co02 6106 co01 6107 coi1 6108 coass 6111 cossxp 6116 fmptco 6883 cofunexg 7639 dftpos4 7900 wunco 10143 relexprelg 14385 relexpaddg 14400 imasless 16801 znleval 20629 metustexhalf 23093 fcoinver 30285 fmptcof2 30330 dfpo2 32888 cnvco1 32892 cnvco2 32893 opelco3 32915 txpss3v 33236 sscoid 33271 xrnss3v 35504 cononrel1 39832 cononrel2 39833 coiun1 39875 relexpaddss 39941 brco2f1o 40260 brco3f1o 40261 neicvgnvor 40344 sblpnf 40519 |
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