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Theorem cosscnvssid3 34561
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 34554 . 2 ( ≀ 𝑅 ⊆ I ↔ ∀𝑥𝑢𝑣((𝑥𝑅𝑢𝑥𝑅𝑣) → 𝑢 = 𝑣))
2 alrot3 2193 . 2 (∀𝑥𝑢𝑣((𝑥𝑅𝑢𝑥𝑅𝑣) → 𝑢 = 𝑣) ↔ ∀𝑢𝑣𝑥((𝑥𝑅𝑢𝑥𝑅𝑣) → 𝑢 = 𝑣))
3 brcnvg 5441 . . . . . 6 ((𝑥 ∈ V ∧ 𝑢 ∈ V) → (𝑥𝑅𝑢𝑢𝑅𝑥))
43el2v 34322 . . . . 5 (𝑥𝑅𝑢𝑢𝑅𝑥)
5 brcnvg 5441 . . . . . 6 ((𝑥 ∈ V ∧ 𝑣 ∈ V) → (𝑥𝑅𝑣𝑣𝑅𝑥))
65el2v 34322 . . . . 5 (𝑥𝑅𝑣𝑣𝑅𝑥)
74, 6anbi12i 604 . . . 4 ((𝑥𝑅𝑢𝑥𝑅𝑣) ↔ (𝑢𝑅𝑥𝑣𝑅𝑥))
87imbi1i 338 . . 3 (((𝑥𝑅𝑢𝑥𝑅𝑣) → 𝑢 = 𝑣) ↔ ((𝑢𝑅𝑥𝑣𝑅𝑥) → 𝑢 = 𝑣))
983albii 34349 . 2 (∀𝑢𝑣𝑥((𝑥𝑅𝑢𝑥𝑅𝑣) → 𝑢 = 𝑣) ↔ ∀𝑢𝑣𝑥((𝑢𝑅𝑥𝑣𝑅𝑥) → 𝑢 = 𝑣))
101, 2, 93bitri 286 1 ( ≀ 𝑅 ⊆ I ↔ ∀𝑢𝑣𝑥((𝑢𝑅𝑥𝑣𝑅𝑥) → 𝑢 = 𝑣))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  wal 1628   = wceq 1630  Vcvv 3349  wss 3721   class class class wbr 4784   I cid 5156  ccnv 5248  ccoss 34308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-br 4785  df-opab 4845  df-id 5157  df-cnv 5257  df-coss 34504
This theorem is referenced by: (None)
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