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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 38424 | . . 3 ⊢ Rel ≀ 𝑅 | |
| 2 | dfcnvrefrel2 38531 | . . 3 ⊢ ( CnvRefRel ≀ 𝑅 ↔ ( ≀ 𝑅 ⊆ ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ∧ Rel ≀ 𝑅)) | |
| 3 | 1, 2 | mpbiran2 710 | . 2 ⊢ ( CnvRefRel ≀ 𝑅 ↔ ≀ 𝑅 ⊆ ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅))) |
| 4 | cossssid 38468 | . 2 ⊢ ( ≀ 𝑅 ⊆ I ↔ ≀ 𝑅 ⊆ ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅))) | |
| 5 | 3, 4 | bitr4i 278 | 1 ⊢ ( CnvRefRel ≀ 𝑅 ↔ ≀ 𝑅 ⊆ I ) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∩ cin 3950 ⊆ wss 3951 I cid 5577 × cxp 5683 dom cdm 5685 ran crn 5686 Rel wrel 5690 ≀ ccoss 38182 CnvRefRel wcnvrefrel 38191 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-coss 38412 df-cnvrefrel 38528 |
| This theorem is referenced by: dffunALTV2 38689 funALTVfun 38699 dfdisjALTV2 38715 |
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