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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 35714 | . 2 ⊢ ( ≀ ◡𝑅 ⊆ I ↔ ∀𝑥∃*𝑢 𝑥◡𝑅𝑢) | |
2 | brcnvg 5753 | . . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑢 ∈ V) → (𝑥◡𝑅𝑢 ↔ 𝑢𝑅𝑥)) | |
3 | 2 | el2v 3504 | . . . 4 ⊢ (𝑥◡𝑅𝑢 ↔ 𝑢𝑅𝑥) |
4 | 3 | mobii 2630 | . . 3 ⊢ (∃*𝑢 𝑥◡𝑅𝑢 ↔ ∃*𝑢 𝑢𝑅𝑥) |
5 | 4 | albii 1819 | . 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 1534 ∃*wmo 2619 Vcvv 3497 ⊆ wss 3939 class class class wbr 5069 I cid 5462 ◡ccnv 5557 ≀ ccoss 35457 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-br 5070 df-opab 5132 df-id 5463 df-cnv 5566 df-coss 35663 |
This theorem is referenced by: cosscnvssid5 35722 dfdisjs4 35948 dfdisjALTV4 35953 eldisjs4 35962 |
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