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Theorem cossssid5 34582
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 34581 . 2 ( ≀ 𝑅 ⊆ I ↔ ∀𝑢∃*𝑥 𝑢𝑅𝑥)
2 ineccnvmo2 34486 . 2 (∀𝑥 ∈ ran 𝑅𝑦 ∈ ran 𝑅(𝑥 = 𝑦 ∨ ([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅) ↔ ∀𝑢∃*𝑥 𝑢𝑅𝑥)
31, 2bitr4i 269 1 ( ≀ 𝑅 ⊆ I ↔ ∀𝑥 ∈ ran 𝑅𝑦 ∈ ran 𝑅(𝑥 = 𝑦 ∨ ([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wb 197  wo 873  wal 1650   = wceq 1652  ∃*wmo 2562  wral 3054  cin 3730  wss 3731  c0 4078   class class class wbr 4808   I cid 5183  ccnv 5275  ran crn 5277  [cec 7944  ccoss 34336
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742  ax-sep 4940  ax-nul 4948  ax-pr 5061
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2564  df-eu 2581  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ral 3059  df-rex 3060  df-rmo 3062  df-rab 3063  df-v 3351  df-sbc 3596  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-nul 4079  df-if 4243  df-sn 4334  df-pr 4336  df-op 4340  df-br 4809  df-opab 4871  df-id 5184  df-xp 5282  df-rel 5283  df-cnv 5284  df-dm 5286  df-rn 5287  df-res 5288  df-ima 5289  df-ec 7948  df-coss 34530
This theorem is referenced by:  cosselcnvrefrels5  34648
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