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Theorem cosscnvssid4 37873
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.)
Assertion
Ref Expression
cosscnvssid4 ( ≀ 𝑅 ⊆ I ↔ ∀𝑥∃*𝑢 𝑢𝑅𝑥)
Distinct variable group:   𝑢,𝑅,𝑥

Proof of Theorem cosscnvssid4
StepHypRef Expression
1 cossssid4 37866 . 2 ( ≀ 𝑅 ⊆ I ↔ ∀𝑥∃*𝑢 𝑥𝑅𝑢)
2 brcnvg 5876 . . . . 5 ((𝑥 ∈ V ∧ 𝑢 ∈ V) → (𝑥𝑅𝑢𝑢𝑅𝑥))
32el2v 3477 . . . 4 (𝑥𝑅𝑢𝑢𝑅𝑥)
43mobii 2537 . . 3 (∃*𝑢 𝑥𝑅𝑢 ↔ ∃*𝑢 𝑢𝑅𝑥)
54albii 1814 . 2 (∀𝑥∃*𝑢 𝑥𝑅𝑢 ↔ ∀𝑥∃*𝑢 𝑢𝑅𝑥)
61, 5bitri 275 1 ( ≀ 𝑅 ⊆ I ↔ ∀𝑥∃*𝑢 𝑢𝑅𝑥)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wal 1532  ∃*wmo 2527  Vcvv 3469  wss 3944   class class class wbr 5142   I cid 5569  ccnv 5671  ccoss 37570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ral 3057  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143  df-opab 5205  df-id 5570  df-cnv 5680  df-coss 37807
This theorem is referenced by:  cosscnvssid5  37874  dfdisjs4  38107  dfdisjALTV4  38112  eldisjs4  38121
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