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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 35712 | . 2 ⊢ ( ≀ ◡𝑅 ⊆ I ↔ ∀𝑥∃*𝑢 𝑥◡𝑅𝑢) | |
2 | brcnvg 5752 | . . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑢 ∈ V) → (𝑥◡𝑅𝑢 ↔ 𝑢𝑅𝑥)) | |
3 | 2 | el2v 3503 | . . . 4 ⊢ (𝑥◡𝑅𝑢 ↔ 𝑢𝑅𝑥) |
4 | 3 | mobii 2631 | . . 3 ⊢ (∃*𝑢 𝑥◡𝑅𝑢 ↔ ∃*𝑢 𝑢𝑅𝑥) |
5 | 4 | albii 1820 | . 2 ⊢ (∀𝑥∃*𝑢 𝑥◡𝑅𝑢 ↔ ∀𝑥∃*𝑢 𝑢𝑅𝑥) |
6 | 1, 5 | bitri 277 | 1 ⊢ ( ≀ ◡𝑅 ⊆ I ↔ ∀𝑥∃*𝑢 𝑢𝑅𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∀wal 1535 ∃*wmo 2620 Vcvv 3496 ⊆ wss 3938 class class class wbr 5068 I cid 5461 ◡ccnv 5556 ≀ ccoss 35455 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-id 5462 df-cnv 5565 df-coss 35661 |
This theorem is referenced by: cosscnvssid5 35720 dfdisjs4 35946 dfdisjALTV4 35951 eldisjs4 35960 |
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