Theorem cossssid 36585

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 36479	. 2 ⊢ ( ≀ 𝑅 ⊆ I ↔ ≀ 𝑅 = ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)))
2		refrelcoss2 36582	. . . 4 ⊢ (( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ⊆ ≀ 𝑅 ∧ Rel ≀ 𝑅)
3	2	simpli 484	. . 3 ⊢ ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ⊆ ≀ 𝑅
4		eqss 3936	. . 3 ⊢ ( ≀ 𝑅 = ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ↔ ( ≀ 𝑅 ⊆ ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ∧ ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ⊆ ≀ 𝑅))
5	3, 4	mpbiran2 707	. 2 ⊢ ( ≀ 𝑅 = ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ↔ ≀ 𝑅 ⊆ ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)))
6	1, 5	bitri 274	1 ⊢ ( ≀ 𝑅 ⊆ I ↔ ≀ 𝑅 ⊆ ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)))

Syntax hints:   ↔ wb 205   = wceq 1539   ∩ cin 3886   ⊆ wss 3887   I cid 5488   × cxp 5587  dom cdm 5589  ran crn 5590  Rel wrel 5594   ≀ ccoss 36333

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-coss 36537

This theorem is referenced by:  cnvrefrelcoss2  36651  cosselcnvrefrels2  36652