Theorem petinidres 40122

Description: A class is a partition by an intersection with the identity class restricted to it if and only if the cosets by the intersection are in equivalence relation on it. Cf. br1cossinidres 39737, disjALTVinidres 40055 and eqvrel1cossinidres 40092. (Contributed by Peter Mazsa, 31-Dec-2021.)

Assertion

Ref	Expression
petinidres	⊢ ((𝑅 ∩ ( I ↾ 𝐴)) Part 𝐴 ↔ ≀ (𝑅 ∩ ( I ↾ 𝐴)) ErALTV 𝐴)

Proof of Theorem petinidres

Step	Hyp	Ref	Expression
1		petinidres2 40121	. 2 ⊢ (( Disj (𝑅 ∩ ( I ↾ 𝐴)) ∧ (dom (𝑅 ∩ ( I ↾ 𝐴)) / (𝑅 ∩ ( I ↾ 𝐴))) = 𝐴) ↔ ( EqvRel ≀ (𝑅 ∩ ( I ↾ 𝐴)) ∧ (dom ≀ (𝑅 ∩ ( I ↾ 𝐴)) / ≀ (𝑅 ∩ ( I ↾ 𝐴))) = 𝐴))
2		dfpart2 40070	. 2 ⊢ ((𝑅 ∩ ( I ↾ 𝐴)) Part 𝐴 ↔ ( Disj (𝑅 ∩ ( I ↾ 𝐴)) ∧ (dom (𝑅 ∩ ( I ↾ 𝐴)) / (𝑅 ∩ ( I ↾ 𝐴))) = 𝐴))
3		dferALTV2 39951	. 2 ⊢ ( ≀ (𝑅 ∩ ( I ↾ 𝐴)) ErALTV 𝐴 ↔ ( EqvRel ≀ (𝑅 ∩ ( I ↾ 𝐴)) ∧ (dom ≀ (𝑅 ∩ ( I ↾ 𝐴)) / ≀ (𝑅 ∩ ( I ↾ 𝐴))) = 𝐴))
4	1, 2, 3	3bitr4i 306	1 ⊢ ((𝑅 ∩ ( I ↾ 𝐴)) Part 𝐴 ↔ ≀ (𝑅 ∩ ( I ↾ 𝐴)) ErALTV 𝐴)

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∩ cin 4092   I cid 5552  dom cdm 5658   ↾ cres 5660   / cqs 8754   ≀ ccoss 39381   EqvRel weqvrel 39398   ErALTV werALTV 39407   Disj wdisjALTV 39417   Part wpart 39422

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 5255  ax-pr 5398

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-rmo 3369  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 4841  df-br 5109  df-opab 5174  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-ec 8757  df-qs 8761  df-coss 39699  df-refrel 39790  df-cnvrefrel 39805  df-symrel 39822  df-trrel 39856  df-eqvrel 39867  df-dmqs 39921  df-erALTV 39947  df-funALTV 39965  df-disjALTV 39988  df-part 40067

This theorem is used by: (None)