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Mirrors > Home > MPE Home > Th. List > Mathboxes > cosselcnvrefrels4 | 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, 31-Aug-2021.) |
Ref | Expression |
---|---|
cosselcnvrefrels4 | ⊢ ( ≀ 𝑅 ∈ CnvRefRels ↔ (∀𝑢∃*𝑥 𝑢𝑅𝑥 ∧ ≀ 𝑅 ∈ Rels )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cosselcnvrefrels2 36295 | . 2 ⊢ ( ≀ 𝑅 ∈ CnvRefRels ↔ ( ≀ 𝑅 ⊆ I ∧ ≀ 𝑅 ∈ Rels )) | |
2 | cossssid4 36231 | . . 3 ⊢ ( ≀ 𝑅 ⊆ I ↔ ∀𝑢∃*𝑥 𝑢𝑅𝑥) | |
3 | 2 | anbi1i 627 | . 2 ⊢ (( ≀ 𝑅 ⊆ I ∧ ≀ 𝑅 ∈ Rels ) ↔ (∀𝑢∃*𝑥 𝑢𝑅𝑥 ∧ ≀ 𝑅 ∈ Rels )) |
4 | 1, 3 | bitri 278 | 1 ⊢ ( ≀ 𝑅 ∈ CnvRefRels ↔ (∀𝑢∃*𝑥 𝑢𝑅𝑥 ∧ ≀ 𝑅 ∈ Rels )) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∀wal 1540 ∈ wcel 2114 ∃*wmo 2538 ⊆ wss 3843 class class class wbr 5030 I cid 5428 ≀ ccoss 35976 Rels crels 35978 CnvRefRels ccnvrefrels 35984 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-coss 36180 df-rels 36246 df-ssr 36259 df-cnvrefs 36284 df-cnvrefrels 36285 |
This theorem is referenced by: dffunsALTV4 36440 elfunsALTV4 36449 |
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