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Mirrors > Home > MPE Home > Th. List > relopab | Structured version Visualization version GIF version |
Description: A class of ordered pairs is a relation. (Contributed by NM, 8-Mar-1995.) Remove disjoint variable conditions. (Revised by Alan Sare, 9-Jul-2013.) (Proof shortened by Mario Carneiro, 21-Dec-2013.) |
Ref | Expression |
---|---|
relopab | ⊢ Rel {〈𝑥, 𝑦〉 ∣ 𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2823 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑥, 𝑦〉 ∣ 𝜑} | |
2 | 1 | relopabi 5696 | 1 ⊢ Rel {〈𝑥, 𝑦〉 ∣ 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: {copab 5130 Rel wrel 5562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-opab 5131 df-xp 5563 df-rel 5564 |
This theorem is referenced by: opabid2 5702 inopab 5703 difopab 5704 dfres2 5911 cnvopab 5999 funopab 6392 relmptopab 7397 elopabi 7762 relmpoopab 7791 shftfn 14434 cicer 17078 joindmss 17619 meetdmss 17633 lgsquadlem3 25960 tgjustf 26261 perpln1 26498 perpln2 26499 fpwrelmapffslem 30470 fpwrelmap 30471 relfae 31508 satfrel 32616 bj-0nelopab 34360 vvdifopab 35523 inxprnres 35551 prtlem12 36005 dicvalrelN 38323 diclspsn 38332 dih1dimatlem 38467 rfovcnvf1od 40357 |
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