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Mirrors > Home > MPE Home > Th. List > brcnvg | Structured version Visualization version GIF version |
Description: The converse of a binary relation swaps arguments. Theorem 11 of [Suppes] p. 61. (Contributed by NM, 10-Oct-2005.) |
Ref | Expression |
---|---|
brcnvg | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴◡𝑅𝐵 ↔ 𝐵𝑅𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq2 5043 | . 2 ⊢ (𝑥 = 𝐴 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝐴)) | |
2 | breq1 5042 | . 2 ⊢ (𝑦 = 𝐵 → (𝑦𝑅𝐴 ↔ 𝐵𝑅𝐴)) | |
3 | df-cnv 5544 | . 2 ⊢ ◡𝑅 = {〈𝑥, 𝑦〉 ∣ 𝑦𝑅𝑥} | |
4 | 1, 2, 3 | brabg 5405 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴◡𝑅𝐵 ↔ 𝐵𝑅𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2112 class class class wbr 5039 ◡ccnv 5535 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-br 5040 df-opab 5102 df-cnv 5544 |
This theorem is referenced by: opelcnvg 5734 brcnv 5736 brelrng 5795 eliniseg 5943 relbrcnvg 5953 brcodir 5964 elpredg 6154 predep 6166 dffv2 6784 ersym 8381 brdifun 8398 eqinf 9078 inflb 9083 infglb 9084 infglbb 9085 infltoreq 9096 infempty 9101 brcnvtrclfv 14531 oduleg 17752 posglbdg 17875 znleval 20473 brbtwn 26944 fcoinvbr 30620 cnvordtrestixx 31531 xrge0iifiso 31553 orvcgteel 32100 fv1stcnv 33421 fv2ndcnv 33422 wsuclem 33499 wsuclb 33502 slenlt 33641 colineardim1 34049 eldmcnv 36166 ineccnvmo 36175 alrmomorn 36176 brxrn 36190 dfcoss3 36226 brcoss3 36242 brcosscnv 36276 cosscnvssid3 36280 cosscnvssid4 36281 brnonrel 40814 ntrneifv2 41308 glbprlem 45875 gte-lte 46040 gt-lt 46041 |
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