Theorem petinidres 38806

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 38434, disjALTVinidres 38742 and eqvrel1cossinidres 38776. (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 38805	. 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 38754	. 2 ⊢ ((𝑅 ∩ ( I ↾ 𝐴)) Part 𝐴 ↔ ( Disj (𝑅 ∩ ( I ↾ 𝐴)) ∧ (dom (𝑅 ∩ ( I ↾ 𝐴)) / (𝑅 ∩ ( I ↾ 𝐴))) = 𝐴))
3		dferALTV2 38653	. 2 ⊢ ( ≀ (𝑅 ∩ ( I ↾ 𝐴)) ErALTV 𝐴 ↔ ( EqvRel ≀ (𝑅 ∩ ( I ↾ 𝐴)) ∧ (dom ≀ (𝑅 ∩ ( I ↾ 𝐴)) / ≀ (𝑅 ∩ ( I ↾ 𝐴))) = 𝐴))
4	1, 2, 3	3bitr4i 303	1 ⊢ ((𝑅 ∩ ( I ↾ 𝐴)) Part 𝐴 ↔ ≀ (𝑅 ∩ ( I ↾ 𝐴)) ErALTV 𝐴)

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∩ cin 3921   I cid 5540  dom cdm 5646   ↾ cres 5648   / cqs 8681   ≀ ccoss 38166   EqvRel weqvrel 38183   ErALTV werALTV 38192   Disj wdisjALTV 38200   Part wpart 38205

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5259  ax-nul 5269  ax-pr 5395

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2880  df-ral 3047  df-rex 3056  df-rmo 3357  df-rab 3412  df-v 3457  df-dif 3925  df-un 3927  df-in 3929  df-ss 3939  df-nul 4305  df-if 4497  df-sn 4598  df-pr 4600  df-op 4604  df-br 5116  df-opab 5178  df-id 5541  df-xp 5652  df-rel 5653  df-cnv 5654  df-co 5655  df-dm 5656  df-rn 5657  df-res 5658  df-ima 5659  df-ec 8684  df-qs 8688  df-coss 38396  df-refrel 38497  df-cnvrefrel 38512  df-symrel 38529  df-trrel 38559  df-eqvrel 38570  df-dmqs 38624  df-erALTV 38649  df-funALTV 38667  df-disjALTV 38690  df-part 38751

This theorem is referenced by: (None)