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Mathbox for Peter Mazsa |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cnvrefrelcoss2 | Structured version Visualization version GIF version |
Description: Necessary and sufficient condition for a coset relation to be a converse reflexive relation. (Contributed by Peter Mazsa, 27-Jul-2021.) |
Ref | Expression |
---|---|
cnvrefrelcoss2 | ⊢ ( CnvRefRel ≀ 𝑅 ↔ ≀ 𝑅 ⊆ I ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relcoss 35220 | . . 3 ⊢ Rel ≀ 𝑅 | |
2 | dfcnvrefrel2 35320 | . . 3 ⊢ ( CnvRefRel ≀ 𝑅 ↔ ( ≀ 𝑅 ⊆ ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ∧ Rel ≀ 𝑅)) | |
3 | 1, 2 | mpbiran2 706 | . 2 ⊢ ( CnvRefRel ≀ 𝑅 ↔ ≀ 𝑅 ⊆ ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅))) |
4 | cossssid 35259 | . 2 ⊢ ( ≀ 𝑅 ⊆ I ↔ ≀ 𝑅 ⊆ ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅))) | |
5 | 3, 4 | bitr4i 279 | 1 ⊢ ( CnvRefRel ≀ 𝑅 ↔ ≀ 𝑅 ⊆ I ) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∩ cin 3864 ⊆ wss 3865 I cid 5354 × cxp 5448 dom cdm 5450 ran crn 5451 Rel wrel 5455 ≀ ccoss 35006 CnvRefRel wcnvrefrel 35015 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-sep 5101 ax-nul 5108 ax-pr 5228 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ral 3112 df-rex 3113 df-rab 3116 df-v 3442 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-nul 4218 df-if 4388 df-sn 4479 df-pr 4481 df-op 4485 df-br 4969 df-opab 5031 df-id 5355 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-coss 35211 df-cnvrefrel 35317 |
This theorem is referenced by: dffunALTV2 35473 funALTVfun 35483 dfdisjALTV2 35499 |
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