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Theorem cossssid4 37998
Description: Equivalent expressions for the class of cosets by 𝑅 to be a subset of the identity class. (Contributed by Peter Mazsa, 31-Aug-2021.)
Assertion
Ref Expression
cossssid4 ( ≀ 𝑅 ⊆ I ↔ ∀𝑢∃*𝑥 𝑢𝑅𝑥)
Distinct variable group:   𝑢,𝑅,𝑥

Proof of Theorem cossssid4
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 cossssid3 37997 . 2 ( ≀ 𝑅 ⊆ I ↔ ∀𝑢𝑥𝑦((𝑢𝑅𝑥𝑢𝑅𝑦) → 𝑥 = 𝑦))
2 breq2 5147 . . . 4 (𝑥 = 𝑦 → (𝑢𝑅𝑥𝑢𝑅𝑦))
32mo4 2554 . . 3 (∃*𝑥 𝑢𝑅𝑥 ↔ ∀𝑥𝑦((𝑢𝑅𝑥𝑢𝑅𝑦) → 𝑥 = 𝑦))
43albii 1813 . 2 (∀𝑢∃*𝑥 𝑢𝑅𝑥 ↔ ∀𝑢𝑥𝑦((𝑢𝑅𝑥𝑢𝑅𝑦) → 𝑥 = 𝑦))
51, 4bitr4i 277 1 ( ≀ 𝑅 ⊆ I ↔ ∀𝑢∃*𝑥 𝑢𝑅𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wal 1531  ∃*wmo 2526  wss 3939   class class class wbr 5143   I cid 5569  ccoss 37705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3052  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5144  df-opab 5206  df-id 5570  df-coss 37939
This theorem is referenced by:  cossssid5  37999  cosscnvssid4  38005  cosselcnvrefrels4  38068  dffunALTV4  38218
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