Theorem eqvrel1cossinidres 39920

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 39883	. 2 ⊢ Disj (𝑅 ∩ ( I ↾ 𝐴))
2	1	disjimi 39911	1 ⊢ EqvRel ≀ (𝑅 ∩ ( I ↾ 𝐴))

This proof depends on syntax axioms:   ∩ cin 4089   I cid 5544   ↾ cres 5652   ≀ ccoss 39209   EqvRel weqvrel 39226

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-coss 39527  df-refrel 39618  df-cnvrefrel 39633  df-symrel 39650  df-trrel 39684  df-eqvrel 39695  df-funALTV 39793  df-disjALTV 39816

This theorem is used by: (None)