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Theorem cossssid5 35751
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 35750 . 2 ( ≀ 𝑅 ⊆ I ↔ ∀𝑢∃*𝑥 𝑢𝑅𝑥)
2 ineccnvmo2 35654 . 2 (∀𝑥 ∈ ran 𝑅𝑦 ∈ ran 𝑅(𝑥 = 𝑦 ∨ ([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅) ↔ ∀𝑢∃*𝑥 𝑢𝑅𝑥)
31, 2bitr4i 281 1 ( ≀ 𝑅 ⊆ I ↔ ∀𝑥 ∈ ran 𝑅𝑦 ∈ ran 𝑅(𝑥 = 𝑦 ∨ ([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wo 844  wal 1536   = wceq 1538  ∃*wmo 2621  wral 3126  cin 3909  wss 3910  c0 4266   class class class wbr 5039   I cid 5432  ccnv 5527  ran crn 5529  [cec 8262  ccoss 35493
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pr 5303
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rmo 3134  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-br 5040  df-opab 5102  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-ec 8266  df-coss 35699
This theorem is referenced by:  cosselcnvrefrels5  35817  dffunALTV5  35964
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