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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 35764 | . 2 ⊢ ( ≀ 𝑅 ∈ CnvRefRels ↔ ( ≀ 𝑅 ⊆ ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ∧ ≀ 𝑅 ∈ Rels )) | |
2 | cossssid 35701 | . . 3 ⊢ ( ≀ 𝑅 ⊆ I ↔ ≀ 𝑅 ⊆ ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅))) | |
3 | 2 | anbi1i 625 | . 2 ⊢ (( ≀ 𝑅 ⊆ I ∧ ≀ 𝑅 ∈ Rels ) ↔ ( ≀ 𝑅 ⊆ ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ∧ ≀ 𝑅 ∈ Rels )) |
4 | 1, 3 | bitr4i 280 | 1 ⊢ ( ≀ 𝑅 ∈ CnvRefRels ↔ ( ≀ 𝑅 ⊆ I ∧ ≀ 𝑅 ∈ Rels )) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∈ wcel 2110 ∩ cin 3934 ⊆ wss 3935 I cid 5453 × cxp 5547 dom cdm 5549 ran crn 5550 ≀ ccoss 35447 Rels crels 35449 CnvRefRels ccnvrefrels 35455 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-coss 35653 df-rels 35719 df-ssr 35732 df-cnvrefs 35757 df-cnvrefrels 35758 |
This theorem is referenced by: cosselcnvrefrels3 35769 cosselcnvrefrels4 35770 cosselcnvrefrels5 35771 dffunsALTV2 35911 elfunsALTV2 35920 dfdisjs2 35936 eldisjs2 35950 |
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