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Theorem cossssid 37337
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
StepHypRef Expression
1 iss2 37213 . 2 ( ≀ 𝑅 ⊆ I ↔ ≀ 𝑅 = ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)))
2 refrelcoss2 37334 . . . 4 (( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ⊆ ≀ 𝑅 ∧ Rel ≀ 𝑅)
32simpli 485 . . 3 ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ⊆ ≀ 𝑅
4 eqss 3998 . . 3 ( ≀ 𝑅 = ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ↔ ( ≀ 𝑅 ⊆ ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ∧ ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ⊆ ≀ 𝑅))
53, 4mpbiran2 709 . 2 ( ≀ 𝑅 = ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ↔ ≀ 𝑅 ⊆ ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)))
61, 5bitri 275 1 ( ≀ 𝑅 ⊆ I ↔ ≀ 𝑅 ⊆ ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1542  cin 3948  wss 3949   I cid 5574   × cxp 5675  dom cdm 5677  ran crn 5678  Rel wrel 5682  ccoss 37043
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-coss 37281
This theorem is referenced by:  cnvrefrelcoss2  37407  cosselcnvrefrels2  37408
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