Theorem cossssid4 38898

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 38897	. 2 ⊢ ( ≀ 𝑅 ⊆ I ↔ ∀𝑢∀𝑥∀𝑦((𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) → 𝑥 = 𝑦))
2		breq2 5090	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑢𝑅𝑥 ↔ 𝑢𝑅𝑦))
3	2	mo4 2567	. . 3 ⊢ (∃*𝑥 𝑢𝑅𝑥 ↔ ∀𝑥∀𝑦((𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) → 𝑥 = 𝑦))
4	3	albii 1821	. 2 ⊢ (∀𝑢∃*𝑥 𝑢𝑅𝑥 ↔ ∀𝑢∀𝑥∀𝑦((𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) → 𝑥 = 𝑦))
5	1, 4	bitr4i 278	1 ⊢ ( ≀ 𝑅 ⊆ I ↔ ∀𝑢∃*𝑥 𝑢𝑅𝑥)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1540  ∃*wmo 2538   ⊆ wss 3890   class class class wbr 5086   I cid 5519   ≀ ccoss 38521

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-id 5520  df-coss 38839

This theorem is referenced by:  cossssid5  38899  cosscnvssid4  38905  cosselcnvrefrels4  38958  dffunALTV4  39113