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Mathbox for Peter Mazsa |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cossssid | Structured version Visualization version GIF version |
Description: Equivalent expressions for the class of cosets by 𝑅 to be a subset of the identity class. (Contributed by Peter Mazsa, 27-Jul-2021.) |
Ref | Expression |
---|---|
cossssid | ⊢ ( ≀ 𝑅 ⊆ I ↔ ≀ 𝑅 ⊆ ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iss2 38326 | . 2 ⊢ ( ≀ 𝑅 ⊆ I ↔ ≀ 𝑅 = ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅))) | |
2 | refrelcoss2 38446 | . . . 4 ⊢ (( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ⊆ ≀ 𝑅 ∧ Rel ≀ 𝑅) | |
3 | 2 | simpli 483 | . . 3 ⊢ ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ⊆ ≀ 𝑅 |
4 | eqss 4011 | . . 3 ⊢ ( ≀ 𝑅 = ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ↔ ( ≀ 𝑅 ⊆ ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ∧ ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ⊆ ≀ 𝑅)) | |
5 | 3, 4 | mpbiran2 710 | . 2 ⊢ ( ≀ 𝑅 = ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ↔ ≀ 𝑅 ⊆ ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅))) |
6 | 1, 5 | bitri 275 | 1 ⊢ ( ≀ 𝑅 ⊆ I ↔ ≀ 𝑅 ⊆ ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 = wceq 1537 ∩ cin 3962 ⊆ wss 3963 I cid 5582 × cxp 5687 dom cdm 5689 ran crn 5690 Rel wrel 5694 ≀ ccoss 38162 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-coss 38393 |
This theorem is referenced by: cnvrefrelcoss2 38519 cosselcnvrefrels2 38520 |
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