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Theorem cosscnvssid4 37342
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 37335 . 2 ( ≀ 𝑅 ⊆ I ↔ ∀𝑥∃*𝑢 𝑥𝑅𝑢)
2 brcnvg 5879 . . . . 5 ((𝑥 ∈ V ∧ 𝑢 ∈ V) → (𝑥𝑅𝑢𝑢𝑅𝑥))
32el2v 3482 . . . 4 (𝑥𝑅𝑢𝑢𝑅𝑥)
43mobii 2542 . . 3 (∃*𝑢 𝑥𝑅𝑢 ↔ ∃*𝑢 𝑢𝑅𝑥)
54albii 1821 . 2 (∀𝑥∃*𝑢 𝑥𝑅𝑢 ↔ ∀𝑥∃*𝑢 𝑢𝑅𝑥)
61, 5bitri 274 1 ( ≀ 𝑅 ⊆ I ↔ ∀𝑥∃*𝑢 𝑢𝑅𝑥)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wal 1539  ∃*wmo 2532  Vcvv 3474  wss 3948   class class class wbr 5148   I cid 5573  ccnv 5675  ccoss 37038
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-id 5574  df-cnv 5684  df-coss 37276
This theorem is referenced by:  cosscnvssid5  37343  dfdisjs4  37576  dfdisjALTV4  37581  eldisjs4  37590
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