Theorem cossssid4 35709

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 35708	. 2 ⊢ ( ≀ 𝑅 ⊆ I ↔ ∀𝑢∀𝑥∀𝑦((𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) → 𝑥 = 𝑦))
2		breq2 5069	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑢𝑅𝑥 ↔ 𝑢𝑅𝑦))
3	2	mo4 2646	. . 3 ⊢ (∃*𝑥 𝑢𝑅𝑥 ↔ ∀𝑥∀𝑦((𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) → 𝑥 = 𝑦))
4	3	albii 1816	. 2 ⊢ (∀𝑢∃*𝑥 𝑢𝑅𝑥 ↔ ∀𝑢∀𝑥∀𝑦((𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) → 𝑥 = 𝑦))
5	1, 4	bitr4i 280	1 ⊢ ( ≀ 𝑅 ⊆ I ↔ ∀𝑢∃*𝑥 𝑢𝑅𝑥)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1531  ∃*wmo 2616   ⊆ wss 3935   class class class wbr 5065   I cid 5458   ≀ ccoss 35452

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pr 5329

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-br 5066  df-opab 5128  df-id 5459  df-coss 35658

This theorem is referenced by:  cossssid5  35710  cosscnvssid4  35716  cosselcnvrefrels4  35775  dffunALTV4  35922