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| Mirrors > Home > MPE Home > Th. List > Mathboxes > cosscnvssid4 | Structured version Visualization version GIF version | ||
| Description: Equivalent expressions for the class of cosets by the converse of 𝑅 to be a subset of the identity class. (Contributed by Peter Mazsa, 31-Aug-2021.) | 
| Ref | Expression | 
|---|---|
| cosscnvssid4 | ⊢ ( ≀ ◡𝑅 ⊆ I ↔ ∀𝑥∃*𝑢 𝑢𝑅𝑥) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | cossssid4 38472 | . 2 ⊢ ( ≀ ◡𝑅 ⊆ I ↔ ∀𝑥∃*𝑢 𝑥◡𝑅𝑢) | |
| 2 | brcnvg 5889 | . . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑢 ∈ V) → (𝑥◡𝑅𝑢 ↔ 𝑢𝑅𝑥)) | |
| 3 | 2 | el2v 3486 | . . . 4 ⊢ (𝑥◡𝑅𝑢 ↔ 𝑢𝑅𝑥) | 
| 4 | 3 | mobii 2547 | . . 3 ⊢ (∃*𝑢 𝑥◡𝑅𝑢 ↔ ∃*𝑢 𝑢𝑅𝑥) | 
| 5 | 4 | albii 1818 | . 2 ⊢ (∀𝑥∃*𝑢 𝑥◡𝑅𝑢 ↔ ∀𝑥∃*𝑢 𝑢𝑅𝑥) | 
| 6 | 1, 5 | bitri 275 | 1 ⊢ ( ≀ ◡𝑅 ⊆ I ↔ ∀𝑥∃*𝑢 𝑢𝑅𝑥) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∀wal 1537 ∃*wmo 2537 Vcvv 3479 ⊆ wss 3950 class class class wbr 5142 I cid 5576 ◡ccnv 5683 ≀ ccoss 38183 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-id 5577 df-cnv 5692 df-coss 38413 | 
| This theorem is referenced by: cosscnvssid5 38480 dfdisjs4 38713 dfdisjALTV4 38718 eldisjs4 38727 | 
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