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Theorem cosscnvssid4 35162
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 35155 . 2 ( ≀ 𝑅 ⊆ I ↔ ∀𝑥∃*𝑢 𝑥𝑅𝑢)
2 brcnvg 5594 . . . . 5 ((𝑥 ∈ V ∧ 𝑢 ∈ V) → (𝑥𝑅𝑢𝑢𝑅𝑥))
32el2v 3416 . . . 4 (𝑥𝑅𝑢𝑢𝑅𝑥)
43mobii 2559 . . 3 (∃*𝑢 𝑥𝑅𝑢 ↔ ∃*𝑢 𝑢𝑅𝑥)
54albii 1782 . 2 (∀𝑥∃*𝑢 𝑥𝑅𝑢 ↔ ∀𝑥∃*𝑢 𝑢𝑅𝑥)
61, 5bitri 267 1 ( ≀ 𝑅 ⊆ I ↔ ∀𝑥∃*𝑢 𝑢𝑅𝑥)
Colors of variables: wff setvar class
Syntax hints:  wb 198  wal 1505  ∃*wmo 2545  Vcvv 3409  wss 3823   class class class wbr 4923   I cid 5305  ccnv 5400  ccoss 34897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-sep 5054  ax-nul 5061  ax-pr 5180
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ral 3087  df-rab 3091  df-v 3411  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-br 4924  df-opab 4986  df-id 5306  df-cnv 5409  df-coss 35104
This theorem is referenced by:  cosscnvssid5  35163  dfdisjs4  35389  dfdisjALTV4  35394  eldisjs4  35403
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