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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 36325 | . 2 ⊢ ( ≀ ◡𝑅 ⊆ I ↔ ∀𝑥∃*𝑢 𝑥◡𝑅𝑢) | |
2 | brcnvg 5748 | . . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑢 ∈ V) → (𝑥◡𝑅𝑢 ↔ 𝑢𝑅𝑥)) | |
3 | 2 | el2v 3416 | . . . 4 ⊢ (𝑥◡𝑅𝑢 ↔ 𝑢𝑅𝑥) |
4 | 3 | mobii 2547 | . . 3 ⊢ (∃*𝑢 𝑥◡𝑅𝑢 ↔ ∃*𝑢 𝑢𝑅𝑥) |
5 | 4 | albii 1827 | . 2 ⊢ (∀𝑥∃*𝑢 𝑥◡𝑅𝑢 ↔ ∀𝑥∃*𝑢 𝑢𝑅𝑥) |
6 | 1, 5 | bitri 278 | 1 ⊢ ( ≀ ◡𝑅 ⊆ I ↔ ∀𝑥∃*𝑢 𝑢𝑅𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∀wal 1541 ∃*wmo 2537 Vcvv 3408 ⊆ wss 3866 class class class wbr 5053 I cid 5454 ◡ccnv 5550 ≀ ccoss 36070 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ral 3066 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-br 5054 df-opab 5116 df-id 5455 df-cnv 5559 df-coss 36274 |
This theorem is referenced by: cosscnvssid5 36333 dfdisjs4 36559 dfdisjALTV4 36564 eldisjs4 36573 |
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