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Theorem cossssid5 39270
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 39269 . 2 ( ≀ 𝑅 ⊆ I ↔ ∀𝑢∃*𝑥 𝑢𝑅𝑥)
2 ineccnvmo2 39077 . 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
This proof depends on syntax axioms:  wb 209  wo 861  wal 1568   = wceq 1570  ∃*wmo 2567  wral 3081  cin 3905  wss 3906  c0 4286   class class class wbr 5111   I cid 5557  ccnv 5662  ran crn 5664  [cec 8698  ccoss 38892
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-pr 5406
This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ral 3082  df-rex 3092  df-rmo 3371  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-ec 8702  df-coss 39210
This theorem is used by:  cosselcnvrefrels5  39330  dffunALTV5  39485
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