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Theorem cosscnvssid4 37285
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 37278 . 2 ( ≀ 𝑅 ⊆ I ↔ ∀𝑥∃*𝑢 𝑥𝑅𝑢)
2 brcnvg 5877 . . . . 5 ((𝑥 ∈ V ∧ 𝑢 ∈ V) → (𝑥𝑅𝑢𝑢𝑅𝑥))
32el2v 3483 . . . 4 (𝑥𝑅𝑢𝑢𝑅𝑥)
43mobii 2543 . . 3 (∃*𝑢 𝑥𝑅𝑢 ↔ ∃*𝑢 𝑢𝑅𝑥)
54albii 1822 . 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 1540  ∃*wmo 2533  Vcvv 3475  wss 3947   class class class wbr 5147   I cid 5572  ccnv 5674  ccoss 36981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-id 5573  df-cnv 5683  df-coss 37219
This theorem is referenced by:  cosscnvssid5  37286  dfdisjs4  37519  dfdisjALTV4  37524  eldisjs4  37533
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