Theorem eqvrel1cossinidres 40029

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 39992	. 2 ⊢ Disj (𝑅 ∩ ( I ↾ 𝐴))
2	1	disjimi 40020	1 ⊢ EqvRel ≀ (𝑅 ∩ ( I ↾ 𝐴))

This proof depends on syntax axioms:   ∩ cin 4092   I cid 5550   ↾ cres 5658   ≀ ccoss 39318   EqvRel weqvrel 39335

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5172  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-coss 39636  df-refrel 39727  df-cnvrefrel 39742  df-symrel 39759  df-trrel 39793  df-eqvrel 39804  df-funALTV 39902  df-disjALTV 39925

This theorem is used by: (None)