Theorem petinidres 39076

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 38708, disjALTVinidres 39012 and eqvrel1cossinidres 39046. (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 39075	. 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 39024	. 2 ⊢ ((𝑅 ∩ ( I ↾ 𝐴)) Part 𝐴 ↔ ( Disj (𝑅 ∩ ( I ↾ 𝐴)) ∧ (dom (𝑅 ∩ ( I ↾ 𝐴)) / (𝑅 ∩ ( I ↾ 𝐴))) = 𝐴))
3		dferALTV2 38923	. 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 1541   ∩ cin 3900   I cid 5518  dom cdm 5624   ↾ cres 5626   / cqs 8634   ≀ ccoss 38379   EqvRel weqvrel 38396   ErALTV werALTV 38405   Disj wdisjALTV 38413   Part wpart 38418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ral 3052  df-rex 3061  df-rmo 3350  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ec 8637  df-qs 8641  df-coss 38670  df-refrel 38761  df-cnvrefrel 38776  df-symrel 38793  df-trrel 38827  df-eqvrel 38838  df-dmqs 38892  df-erALTV 38919  df-funALTV 38937  df-disjALTV 38960  df-part 39021

This theorem is referenced by: (None)