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Theorem cossssid5 38169
Description: Equivalent expressions for the class of cosets by 𝑅 to be a subset of the identity class. (Contributed by Peter Mazsa, 5-Sep-2021.)
Assertion
Ref Expression
cossssid5 ( ≀ 𝑅 ⊆ I ↔ ∀𝑥 ∈ ran 𝑅𝑦 ∈ ran 𝑅(𝑥 = 𝑦 ∨ ([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅))
Distinct variable group:   𝑥,𝑅,𝑦

Proof of Theorem cossssid5
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 cossssid4 38168 . 2 ( ≀ 𝑅 ⊆ I ↔ ∀𝑢∃*𝑥 𝑢𝑅𝑥)
2 ineccnvmo2 38058 . 2 (∀𝑥 ∈ ran 𝑅𝑦 ∈ ran 𝑅(𝑥 = 𝑦 ∨ ([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅) ↔ ∀𝑢∃*𝑥 𝑢𝑅𝑥)
31, 2bitr4i 277 1 ( ≀ 𝑅 ⊆ I ↔ ∀𝑥 ∈ ran 𝑅𝑦 ∈ ran 𝑅(𝑥 = 𝑦 ∨ ([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wo 845  wal 1532   = wceq 1534  ∃*wmo 2527  wral 3051  cin 3946  wss 3947  c0 4325   class class class wbr 5153   I cid 5579  ccnv 5681  ran crn 5683  [cec 8732  ccoss 37876
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 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ral 3052  df-rex 3061  df-rmo 3364  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-br 5154  df-opab 5216  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-ec 8736  df-coss 38109
This theorem is referenced by:  cosselcnvrefrels5  38239  dffunALTV5  38389
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