Theorem cossssid4 39229

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.

Step	Hyp	Ref	Expression
1		cossssid3 39228	. 2 ⊢ ( ≀ 𝑅 ⊆ I ↔ ∀𝑢∀𝑥∀𝑦((𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) → 𝑥 = 𝑦))
2		breq2 5113	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑢𝑅𝑥 ↔ 𝑢𝑅𝑦))
3	2	mo4 2594	. . 3 ⊢ (∃*𝑥 𝑢𝑅𝑥 ↔ ∀𝑥∀𝑦((𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) → 𝑥 = 𝑦))
4	3	albii 1849	. 2 ⊢ (∀𝑢∃*𝑥 𝑢𝑅𝑥 ↔ ∀𝑢∀𝑥∀𝑦((𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) → 𝑥 = 𝑦))
5	1, 4	bitr4i 281	1 ⊢ ( ≀ 𝑅 ⊆ I ↔ ∀𝑢∃*𝑥 𝑢𝑅𝑥)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1568  ∃*wmo 2565   ⊆ wss 3905   class class class wbr 5109   I cid 5555   ≀ ccoss 38852

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-id 5556  df-coss 39170

This theorem is referenced by:  cossssid5  39230  cosscnvssid4  39236  cosselcnvrefrels4  39289  dffunALTV4  39444