![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > brcnv | GIF version |
Description: The converse of a binary relation swaps arguments. Theorem 11 of [Suppes] p. 61. (Contributed by NM, 13-Aug-1995.) |
Ref | Expression |
---|---|
opelcnv.1 | ⊢ 𝐴 ∈ V |
opelcnv.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
brcnv | ⊢ (𝐴◡𝑅𝐵 ↔ 𝐵𝑅𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelcnv.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | opelcnv.2 | . 2 ⊢ 𝐵 ∈ V | |
3 | brcnvg 4808 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴◡𝑅𝐵 ↔ 𝐵𝑅𝐴)) | |
4 | 1, 2, 3 | mp2an 426 | 1 ⊢ (𝐴◡𝑅𝐵 ↔ 𝐵𝑅𝐴) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 ∈ wcel 2148 Vcvv 2737 class class class wbr 4003 ◡ccnv 4625 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-br 4004 df-opab 4065 df-cnv 4634 |
This theorem is referenced by: cnvco 4812 dfrn2 4815 dfdm4 4819 cnvsym 5012 intasym 5013 asymref 5014 qfto 5018 dminss 5043 imainss 5044 dminxp 5073 cnvcnv3 5078 cnvpom 5171 cnvsom 5172 dffun2 5226 funcnvsn 5261 funcnv2 5276 funcnveq 5279 fun2cnv 5280 imadif 5296 f1ompt 5667 f1eqcocnv 5791 fliftcnv 5795 isocnv2 5812 ercnv 6555 ecid 6597 cnvinfex 7016 eqinfti 7018 infvalti 7020 infmoti 7026 dfinfre 8912 pw1nct 14688 |
Copyright terms: Public domain | W3C validator |