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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 36823 | . . 3 ⊢ Rel ≀ 𝑅 | |
2 | dfcnvrefrel2 36930 | . . 3 ⊢ ( CnvRefRel ≀ 𝑅 ↔ ( ≀ 𝑅 ⊆ ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ∧ Rel ≀ 𝑅)) | |
3 | 1, 2 | mpbiran2 708 | . 2 ⊢ ( CnvRefRel ≀ 𝑅 ↔ ≀ 𝑅 ⊆ ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅))) |
4 | cossssid 36867 | . 2 ⊢ ( ≀ 𝑅 ⊆ I ↔ ≀ 𝑅 ⊆ ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅))) | |
5 | 3, 4 | bitr4i 277 | 1 ⊢ ( CnvRefRel ≀ 𝑅 ↔ ≀ 𝑅 ⊆ I ) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∩ cin 3907 ⊆ wss 3908 I cid 5528 × cxp 5629 dom cdm 5631 ran crn 5632 Rel wrel 5636 ≀ ccoss 36572 CnvRefRel wcnvrefrel 36581 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pr 5382 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-br 5104 df-opab 5166 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-coss 36811 df-cnvrefrel 36927 |
This theorem is referenced by: dffunALTV2 37088 funALTVfun 37098 dfdisjALTV2 37114 |
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