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Theorem cossssid4 38452
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 38451 . 2 ( ≀ 𝑅 ⊆ I ↔ ∀𝑢𝑥𝑦((𝑢𝑅𝑥𝑢𝑅𝑦) → 𝑥 = 𝑦))
2 breq2 5152 . . . 4 (𝑥 = 𝑦 → (𝑢𝑅𝑥𝑢𝑅𝑦))
32mo4 2564 . . 3 (∃*𝑥 𝑢𝑅𝑥 ↔ ∀𝑥𝑦((𝑢𝑅𝑥𝑢𝑅𝑦) → 𝑥 = 𝑦))
43albii 1816 . 2 (∀𝑢∃*𝑥 𝑢𝑅𝑥 ↔ ∀𝑢𝑥𝑦((𝑢𝑅𝑥𝑢𝑅𝑦) → 𝑥 = 𝑦))
51, 4bitr4i 278 1 ( ≀ 𝑅 ⊆ I ↔ ∀𝑢∃*𝑥 𝑢𝑅𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1535  ∃*wmo 2536  wss 3963   class class class wbr 5148   I cid 5582  ccoss 38162
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-coss 38393
This theorem is referenced by:  cossssid5  38453  cosscnvssid4  38459  cosselcnvrefrels4  38522  dffunALTV4  38672
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