Theorem petidres 37689

Description: A class is a partition by identity class restricted to it if and only if the cosets by the restricted identity class are in equivalence relation on it, cf. eqvrel1cossidres 37660. (Contributed by Peter Mazsa, 31-Dec-2021.)

Assertion

Ref	Expression
petidres	⊢ (( I ↾ 𝐴) Part 𝐴 ↔ ≀ ( I ↾ 𝐴) ErALTV 𝐴)

Proof of Theorem petidres

Step	Hyp	Ref	Expression
1		petidres2 37688	. 2 ⊢ (( Disj ( I ↾ 𝐴) ∧ (dom ( I ↾ 𝐴) / ( I ↾ 𝐴)) = 𝐴) ↔ ( EqvRel ≀ ( I ↾ 𝐴) ∧ (dom ≀ ( I ↾ 𝐴) / ≀ ( I ↾ 𝐴)) = 𝐴))
2		dfpart2 37639	. 2 ⊢ (( I ↾ 𝐴) Part 𝐴 ↔ ( Disj ( I ↾ 𝐴) ∧ (dom ( I ↾ 𝐴) / ( I ↾ 𝐴)) = 𝐴))
3		dferALTV2 37538	. 2 ⊢ ( ≀ ( I ↾ 𝐴) ErALTV 𝐴 ↔ ( EqvRel ≀ ( I ↾ 𝐴) ∧ (dom ≀ ( I ↾ 𝐴) / ≀ ( I ↾ 𝐴)) = 𝐴))
4	1, 2, 3	3bitr4i 303	1 ⊢ (( I ↾ 𝐴) Part 𝐴 ↔ ≀ ( I ↾ 𝐴) ErALTV 𝐴)

Syntax hints:   ↔ wb 205   ∧ wa 397   = wceq 1542   I cid 5574  dom cdm 5677   ↾ cres 5679   / cqs 8702   ≀ ccoss 37043   EqvRel weqvrel 37060   ErALTV werALTV 37069   Disj wdisjALTV 37077   Part wpart 37082

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rmo 3377  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ec 8705  df-qs 8709  df-coss 37281  df-refrel 37382  df-cnvrefrel 37397  df-symrel 37414  df-trrel 37444  df-eqvrel 37455  df-dmqs 37509  df-erALTV 37534  df-funALTV 37552  df-disjALTV 37575  df-part 37636

This theorem is referenced by: (None)