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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 35654 | . 2 ⊢ ( ≀ 𝑅 ∈ CnvRefRels ↔ ( ≀ 𝑅 ⊆ I ∧ ≀ 𝑅 ∈ Rels )) | |
2 | cossssid4 35590 | . . 3 ⊢ ( ≀ 𝑅 ⊆ I ↔ ∀𝑢∃*𝑥 𝑢𝑅𝑥) | |
3 | 2 | anbi1i 623 | . 2 ⊢ (( ≀ 𝑅 ⊆ I ∧ ≀ 𝑅 ∈ Rels ) ↔ (∀𝑢∃*𝑥 𝑢𝑅𝑥 ∧ ≀ 𝑅 ∈ Rels )) |
4 | 1, 3 | bitri 276 | 1 ⊢ ( ≀ 𝑅 ∈ CnvRefRels ↔ (∀𝑢∃*𝑥 𝑢𝑅𝑥 ∧ ≀ 𝑅 ∈ Rels )) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 ∀wal 1526 ∈ wcel 2105 ∃*wmo 2613 ⊆ wss 3933 class class class wbr 5057 I cid 5452 ≀ ccoss 35334 Rels crels 35336 CnvRefRels ccnvrefrels 35342 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-coss 35539 df-rels 35605 df-ssr 35618 df-cnvrefs 35643 df-cnvrefrels 35644 |
This theorem is referenced by: dffunsALTV4 35799 elfunsALTV4 35808 |
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