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Mirrors > Home > MPE Home > Th. List > Mathboxes > refrelcoss | Structured version Visualization version GIF version |
Description: The class of cosets by 𝑅 is reflexive. (Contributed by Peter Mazsa, 4-Jul-2020.) |
Ref | Expression |
---|---|
refrelcoss | ⊢ RefRel ≀ 𝑅 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | refrelcoss2 35248 | . 2 ⊢ (( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ⊆ ≀ 𝑅 ∧ Rel ≀ 𝑅) | |
2 | dfrefrel2 35299 | . 2 ⊢ ( RefRel ≀ 𝑅 ↔ (( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ⊆ ≀ 𝑅 ∧ Rel ≀ 𝑅)) | |
3 | 1, 2 | mpbir 232 | 1 ⊢ RefRel ≀ 𝑅 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 ∩ cin 3860 ⊆ wss 3861 I cid 5350 × cxp 5444 dom cdm 5446 ran crn 5447 Rel wrel 5451 ≀ ccoss 34998 RefRel wrefrel 35004 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1778 ax-4 1792 ax-5 1889 ax-6 1948 ax-7 1993 ax-8 2082 ax-9 2090 ax-10 2111 ax-11 2125 ax-12 2140 ax-13 2343 ax-ext 2768 ax-sep 5097 ax-nul 5104 ax-pr 5224 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1763 df-nf 1767 df-sb 2042 df-mo 2575 df-eu 2611 df-clab 2775 df-cleq 2787 df-clel 2862 df-nfc 2934 df-ral 3109 df-rex 3110 df-rab 3113 df-v 3438 df-dif 3864 df-un 3866 df-in 3868 df-ss 3876 df-nul 4214 df-if 4384 df-sn 4475 df-pr 4477 df-op 4481 df-br 4965 df-opab 5027 df-id 5351 df-xp 5452 df-rel 5453 df-cnv 5454 df-co 5455 df-dm 5456 df-rn 5457 df-res 5458 df-coss 35203 df-refrel 35296 |
This theorem is referenced by: eqvrelcoss 35396 |
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