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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 5321 | . 2 ⊢ (𝐴 ∘ 𝐵) = {〈𝑥, 𝑦〉 ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)} | |
2 | 1 | relopabi 5449 | 1 ⊢ Rel (𝐴 ∘ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 385 ∃wex 1875 class class class wbr 4843 ∘ ccom 5316 Rel wrel 5317 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-rab 3098 df-v 3387 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-sn 4369 df-pr 4371 df-op 4375 df-opab 4906 df-xp 5318 df-rel 5319 df-co 5321 |
This theorem is referenced by: dfco2 5853 resco 5858 coeq0 5863 coiun 5864 cocnvcnv2 5866 cores2 5867 co02 5868 co01 5869 coi1 5870 coass 5873 cossxp 5877 fmptco 6623 cofunexg 7365 dftpos4 7609 wunco 9843 relexprelg 14119 relexpaddg 14134 imasless 16515 znleval 20224 metustexhalf 22689 fcoinver 29935 fmptcof2 29976 dfpo2 32159 cnvco1 32163 cnvco2 32164 opelco3 32190 txpss3v 32498 sscoid 32533 xrnss3v 34628 cononrel1 38683 cononrel2 38684 coiun1 38727 relexpaddss 38793 brco2f1o 39112 brco3f1o 39113 neicvgnvor 39196 sblpnf 39291 |
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