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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 5070 | . 2 ⊢ (𝑥 = 𝐴 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝐴)) | |
2 | breq1 5069 | . 2 ⊢ (𝑦 = 𝐵 → (𝑦𝑅𝐴 ↔ 𝐵𝑅𝐴)) | |
3 | df-cnv 5563 | . 2 ⊢ ◡𝑅 = {〈𝑥, 𝑦〉 ∣ 𝑦𝑅𝑥} | |
4 | 1, 2, 3 | brabg 5426 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴◡𝑅𝐵 ↔ 𝐵𝑅𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2114 class class class wbr 5066 ◡ccnv 5554 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
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-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-cnv 5563 |
This theorem is referenced by: opelcnvg 5751 brcnv 5753 brelrng 5811 eliniseg 5958 relbrcnvg 5968 brcodir 5979 elpredg 6162 predep 6174 dffv2 6756 ersym 8301 brdifun 8318 eqinf 8948 inflb 8953 infglb 8954 infglbb 8955 infltoreq 8966 infempty 8971 brcnvtrclfv 14363 oduleg 17742 posglbd 17760 znleval 20701 brbtwn 26685 fcoinvbr 30358 cnvordtrestixx 31156 xrge0iifiso 31178 orvcgteel 31725 fv1stcnv 33020 fv2ndcnv 33021 wsuclem 33112 wsuclb 33115 slenlt 33231 colineardim1 33522 eldmcnv 35617 ineccnvmo 35626 alrmomorn 35627 brxrn 35641 dfcoss3 35677 brcoss3 35693 brcosscnv 35727 cosscnvssid3 35731 cosscnvssid4 35732 brnonrel 39969 ntrneifv2 40450 gte-lte 44843 gt-lt 44844 |
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