Theorem cossssid4 38471

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 38470	. 2 ⊢ ( ≀ 𝑅 ⊆ I ↔ ∀𝑢∀𝑥∀𝑦((𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) → 𝑥 = 𝑦))
2		breq2 5147	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑢𝑅𝑥 ↔ 𝑢𝑅𝑦))
3	2	mo4 2566	. . 3 ⊢ (∃*𝑥 𝑢𝑅𝑥 ↔ ∀𝑥∀𝑦((𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) → 𝑥 = 𝑦))
4	3	albii 1819	. 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 1538  ∃*wmo 2538   ⊆ wss 3951   class class class wbr 5143   I cid 5577   ≀ ccoss 38182

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-coss 38412

This theorem is referenced by:  cossssid5  38472  cosscnvssid4  38478  cosselcnvrefrels4  38541  dffunALTV4  38691