Theorem cossssid 38458

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 38326	. 2 ⊢ ( ≀ 𝑅 ⊆ I ↔ ≀ 𝑅 = ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)))
2		refrelcoss2 38455	. . . 4 ⊢ (( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ⊆ ≀ 𝑅 ∧ Rel ≀ 𝑅)
3	2	simpli 483	. . 3 ⊢ ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ⊆ ≀ 𝑅
4		eqss 3962	. . 3 ⊢ ( ≀ 𝑅 = ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ↔ ( ≀ 𝑅 ⊆ ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ∧ ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ⊆ ≀ 𝑅))
5	3, 4	mpbiran2 710	. 2 ⊢ ( ≀ 𝑅 = ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ↔ ≀ 𝑅 ⊆ ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)))
6	1, 5	bitri 275	1 ⊢ ( ≀ 𝑅 ⊆ I ↔ ≀ 𝑅 ⊆ ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)))

Syntax hints:   ↔ wb 206   = wceq 1540   ∩ cin 3913   ⊆ wss 3914   I cid 5532   × cxp 5636  dom cdm 5638  ran crn 5639  Rel wrel 5643   ≀ ccoss 38169

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-coss 38402

This theorem is referenced by:  cnvrefrelcoss2  38528  cosselcnvrefrels2  38529