Theorem cossssid 36169

Description: Equivalent expressions for the class of cosets by 𝑅 to be a subset of the identity class. (Contributed by Peter Mazsa, 27-Jul-2021.)

Assertion

Ref	Expression
cossssid	⊢ ( ≀ 𝑅 ⊆ I ↔ ≀ 𝑅 ⊆ ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)))

Proof of Theorem cossssid

Step	Hyp	Ref	Expression
1		iss2 36063	. 2 ⊢ ( ≀ 𝑅 ⊆ I ↔ ≀ 𝑅 = ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)))
2		refrelcoss2 36166	. . . 4 ⊢ (( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ⊆ ≀ 𝑅 ∧ Rel ≀ 𝑅)
3	2	simpli 487	. . 3 ⊢ ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ⊆ ≀ 𝑅
4		eqss 3907	. . 3 ⊢ ( ≀ 𝑅 = ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ↔ ( ≀ 𝑅 ⊆ ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ∧ ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ⊆ ≀ 𝑅))
5	3, 4	mpbiran2 709	. 2 ⊢ ( ≀ 𝑅 = ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ↔ ≀ 𝑅 ⊆ ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)))
6	1, 5	bitri 278	1 ⊢ ( ≀ 𝑅 ⊆ I ↔ ≀ 𝑅 ⊆ ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)))

Syntax hints:   ↔ wb 209   = wceq 1538   ∩ cin 3857   ⊆ wss 3858   I cid 5429   × cxp 5522  dom cdm 5524  ran crn 5525  Rel wrel 5529   ≀ ccoss 35915

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-br 5033  df-opab 5095  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-coss 36121

This theorem is referenced by:  cnvrefrelcoss2  36235  cosselcnvrefrels2  36236