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Theorem cosscnvssid3 34536
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.)
Assertion
Ref Expression
cosscnvssid3 ( ≀ 𝑅 ⊆ I ↔ ∀𝑢𝑣𝑥((𝑢𝑅𝑥𝑣𝑅𝑥) → 𝑢 = 𝑣))
Distinct variable group:   𝑢,𝑅,𝑣,𝑥

Proof of Theorem cosscnvssid3
StepHypRef Expression
1 cossssid3 34529 . 2 ( ≀ 𝑅 ⊆ I ↔ ∀𝑥𝑢𝑣((𝑥𝑅𝑢𝑥𝑅𝑣) → 𝑢 = 𝑣))
2 alrot3 2207 . 2 (∀𝑥𝑢𝑣((𝑥𝑅𝑢𝑥𝑅𝑣) → 𝑢 = 𝑣) ↔ ∀𝑢𝑣𝑥((𝑥𝑅𝑢𝑥𝑅𝑣) → 𝑢 = 𝑣))
3 brcnvg 5501 . . . . . 6 ((𝑥 ∈ V ∧ 𝑢 ∈ V) → (𝑥𝑅𝑢𝑢𝑅𝑥))
43el2v 34303 . . . . 5 (𝑥𝑅𝑢𝑢𝑅𝑥)
5 brcnvg 5501 . . . . . 6 ((𝑥 ∈ V ∧ 𝑣 ∈ V) → (𝑥𝑅𝑣𝑣𝑅𝑥))
65el2v 34303 . . . . 5 (𝑥𝑅𝑣𝑣𝑅𝑥)
74, 6anbi12i 614 . . . 4 ((𝑥𝑅𝑢𝑥𝑅𝑣) ↔ (𝑢𝑅𝑥𝑣𝑅𝑥))
87imbi1i 340 . . 3 (((𝑥𝑅𝑢𝑥𝑅𝑣) → 𝑢 = 𝑣) ↔ ((𝑢𝑅𝑥𝑣𝑅𝑥) → 𝑢 = 𝑣))
983albii 34328 . 2 (∀𝑢𝑣𝑥((𝑥𝑅𝑢𝑥𝑅𝑣) → 𝑢 = 𝑣) ↔ ∀𝑢𝑣𝑥((𝑢𝑅𝑥𝑣𝑅𝑥) → 𝑢 = 𝑣))
101, 2, 93bitri 288 1 ( ≀ 𝑅 ⊆ I ↔ ∀𝑢𝑣𝑥((𝑢𝑅𝑥𝑣𝑅𝑥) → 𝑢 = 𝑣))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wal 1635   = wceq 1637  Vcvv 3390  wss 3766   class class class wbr 4840   I cid 5215  ccnv 5307  ccoss 34290
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1880  ax-4 1897  ax-5 2004  ax-6 2070  ax-7 2106  ax-9 2167  ax-10 2187  ax-11 2203  ax-12 2216  ax-13 2422  ax-ext 2784  ax-sep 4971  ax-nul 4980  ax-pr 5093
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1865  df-sb 2063  df-eu 2636  df-mo 2637  df-clab 2792  df-cleq 2798  df-clel 2801  df-nfc 2936  df-ral 3100  df-rab 3104  df-v 3392  df-dif 3769  df-un 3771  df-in 3773  df-ss 3780  df-nul 4114  df-if 4277  df-sn 4368  df-pr 4370  df-op 4374  df-br 4841  df-opab 4903  df-id 5216  df-cnv 5316  df-coss 34479
This theorem is referenced by: (None)
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