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Mathbox for Peter Mazsa |
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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 35155 | . 2 ⊢ ( ≀ ◡𝑅 ⊆ I ↔ ∀𝑥∃*𝑢 𝑥◡𝑅𝑢) | |
2 | brcnvg 5594 | . . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑢 ∈ V) → (𝑥◡𝑅𝑢 ↔ 𝑢𝑅𝑥)) | |
3 | 2 | el2v 3416 | . . . 4 ⊢ (𝑥◡𝑅𝑢 ↔ 𝑢𝑅𝑥) |
4 | 3 | mobii 2559 | . . 3 ⊢ (∃*𝑢 𝑥◡𝑅𝑢 ↔ ∃*𝑢 𝑢𝑅𝑥) |
5 | 4 | albii 1782 | . 2 ⊢ (∀𝑥∃*𝑢 𝑥◡𝑅𝑢 ↔ ∀𝑥∃*𝑢 𝑢𝑅𝑥) |
6 | 1, 5 | bitri 267 | 1 ⊢ ( ≀ ◡𝑅 ⊆ I ↔ ∀𝑥∃*𝑢 𝑢𝑅𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∀wal 1505 ∃*wmo 2545 Vcvv 3409 ⊆ wss 3823 class class class wbr 4923 I cid 5305 ◡ccnv 5400 ≀ ccoss 34897 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-sep 5054 ax-nul 5061 ax-pr 5180 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ral 3087 df-rab 3091 df-v 3411 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-nul 4173 df-if 4345 df-sn 4436 df-pr 4438 df-op 4442 df-br 4924 df-opab 4986 df-id 5306 df-cnv 5409 df-coss 35104 |
This theorem is referenced by: cosscnvssid5 35163 dfdisjs4 35389 dfdisjALTV4 35394 eldisjs4 35403 |
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