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Mirrors > Home > MPE Home > Th. List > Mathboxes > cosscnvssid3 | 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, 28-Jul-2021.) |
Ref | Expression |
---|---|
cosscnvssid3 | ⊢ ( ≀ ◡𝑅 ⊆ I ↔ ∀𝑢∀𝑣∀𝑥((𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥) → 𝑢 = 𝑣)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cossssid3 35589 | . 2 ⊢ ( ≀ ◡𝑅 ⊆ I ↔ ∀𝑥∀𝑢∀𝑣((𝑥◡𝑅𝑢 ∧ 𝑥◡𝑅𝑣) → 𝑢 = 𝑣)) | |
2 | alrot3 2154 | . 2 ⊢ (∀𝑥∀𝑢∀𝑣((𝑥◡𝑅𝑢 ∧ 𝑥◡𝑅𝑣) → 𝑢 = 𝑣) ↔ ∀𝑢∀𝑣∀𝑥((𝑥◡𝑅𝑢 ∧ 𝑥◡𝑅𝑣) → 𝑢 = 𝑣)) | |
3 | brcnvg 5743 | . . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑢 ∈ V) → (𝑥◡𝑅𝑢 ↔ 𝑢𝑅𝑥)) | |
4 | 3 | el2v 3499 | . . . . 5 ⊢ (𝑥◡𝑅𝑢 ↔ 𝑢𝑅𝑥) |
5 | brcnvg 5743 | . . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑣 ∈ V) → (𝑥◡𝑅𝑣 ↔ 𝑣𝑅𝑥)) | |
6 | 5 | el2v 3499 | . . . . 5 ⊢ (𝑥◡𝑅𝑣 ↔ 𝑣𝑅𝑥) |
7 | 4, 6 | anbi12i 626 | . . . 4 ⊢ ((𝑥◡𝑅𝑢 ∧ 𝑥◡𝑅𝑣) ↔ (𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥)) |
8 | 7 | imbi1i 351 | . . 3 ⊢ (((𝑥◡𝑅𝑢 ∧ 𝑥◡𝑅𝑣) → 𝑢 = 𝑣) ↔ ((𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥) → 𝑢 = 𝑣)) |
9 | 8 | 3albii 35390 | . 2 ⊢ (∀𝑢∀𝑣∀𝑥((𝑥◡𝑅𝑢 ∧ 𝑥◡𝑅𝑣) → 𝑢 = 𝑣) ↔ ∀𝑢∀𝑣∀𝑥((𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥) → 𝑢 = 𝑣)) |
10 | 1, 2, 9 | 3bitri 298 | 1 ⊢ ( ≀ ◡𝑅 ⊆ I ↔ ∀𝑢∀𝑣∀𝑥((𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥) → 𝑢 = 𝑣)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∀wal 1526 Vcvv 3492 ⊆ wss 3933 class class class wbr 5057 I cid 5452 ◡ccnv 5547 ≀ ccoss 35334 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-id 5453 df-cnv 5556 df-coss 35539 |
This theorem is referenced by: dfdisjs3 35823 dfdisjALTV3 35828 eldisjs3 35837 |
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