Theorem eqvrel1cossinidres 37661

Description: The cosets by an intersection with a restricted identity relation are in equivalence relation. (Contributed by Peter Mazsa, 31-Dec-2021.)

Assertion

Ref	Expression
eqvrel1cossinidres	⊢ EqvRel ≀ (𝑅 ∩ ( I ↾ 𝐴))

Proof of Theorem eqvrel1cossinidres

Step	Hyp	Ref	Expression
1		disjALTVinidres 37627	. 2 ⊢ Disj (𝑅 ∩ ( I ↾ 𝐴))
2	1	disjimi 37652	1 ⊢ EqvRel ≀ (𝑅 ∩ ( I ↾ 𝐴))

Syntax hints:   ∩ cin 3948   I cid 5574   ↾ cres 5679   ≀ ccoss 37043   EqvRel weqvrel 37060

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-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  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  df-refrel 37382  df-cnvrefrel 37397  df-symrel 37414  df-trrel 37444  df-eqvrel 37455  df-funALTV 37552  df-disjALTV 37575

This theorem is referenced by: (None)