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Theorem cosscnvssid4 38459
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 38452 . 2 ( ≀ 𝑅 ⊆ I ↔ ∀𝑥∃*𝑢 𝑥𝑅𝑢)
2 brcnvg 5893 . . . . 5 ((𝑥 ∈ V ∧ 𝑢 ∈ V) → (𝑥𝑅𝑢𝑢𝑅𝑥))
32el2v 3485 . . . 4 (𝑥𝑅𝑢𝑢𝑅𝑥)
43mobii 2546 . . 3 (∃*𝑢 𝑥𝑅𝑢 ↔ ∃*𝑢 𝑢𝑅𝑥)
54albii 1816 . 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 206  wal 1535  ∃*wmo 2536  Vcvv 3478  wss 3963   class class class wbr 5148   I cid 5582  ccnv 5688  ccoss 38162
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-cnv 5697  df-coss 38393
This theorem is referenced by:  cosscnvssid5  38460  dfdisjs4  38693  dfdisjALTV4  38698  eldisjs4  38707
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