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| Mirrors > Home > MPE Home > Th. List > Mathboxes > cosselcnvrefrels2 | Structured version Visualization version GIF version | ||
| Description: Necessary and sufficient condition for a coset relation to be an element of the converse reflexive relation class. (Contributed by Peter Mazsa, 25-Aug-2021.) |
| Ref | Expression |
|---|---|
| cosselcnvrefrels2 | ⊢ ( ≀ 𝑅 ∈ CnvRefRels ↔ ( ≀ 𝑅 ⊆ I ∧ ≀ 𝑅 ∈ Rels )) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elcnvrefrels2 38535 | . 2 ⊢ ( ≀ 𝑅 ∈ CnvRefRels ↔ ( ≀ 𝑅 ⊆ ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ∧ ≀ 𝑅 ∈ Rels )) | |
| 2 | cossssid 38468 | . . 3 ⊢ ( ≀ 𝑅 ⊆ I ↔ ≀ 𝑅 ⊆ ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅))) | |
| 3 | 2 | anbi1i 624 | . 2 ⊢ (( ≀ 𝑅 ⊆ I ∧ ≀ 𝑅 ∈ Rels ) ↔ ( ≀ 𝑅 ⊆ ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ∧ ≀ 𝑅 ∈ Rels )) |
| 4 | 1, 3 | bitr4i 278 | 1 ⊢ ( ≀ 𝑅 ∈ CnvRefRels ↔ ( ≀ 𝑅 ⊆ I ∧ ≀ 𝑅 ∈ Rels )) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 ∈ wcel 2108 ∩ cin 3950 ⊆ wss 3951 I cid 5577 × cxp 5683 dom cdm 5685 ran crn 5686 ≀ ccoss 38182 Rels crels 38184 CnvRefRels ccnvrefrels 38190 |
| 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-pow 5365 ax-pr 5432 ax-un 7755 |
| 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-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 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-rels 38486 df-ssr 38499 df-cnvrefs 38526 df-cnvrefrels 38527 |
| This theorem is referenced by: cosselcnvrefrels3 38540 cosselcnvrefrels4 38541 cosselcnvrefrels5 38542 dffunsALTV2 38685 elfunsALTV2 38694 dfdisjs2 38710 eldisjs2 38724 |
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