Theorem petidres2 39124

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

Assertion

Ref	Expression
petidres2	⊢ (( Disj ( I ↾ 𝐴) ∧ (dom ( I ↾ 𝐴) / ( I ↾ 𝐴)) = 𝐴) ↔ ( EqvRel ≀ ( I ↾ 𝐴) ∧ (dom ≀ ( I ↾ 𝐴) / ≀ ( I ↾ 𝐴)) = 𝐴))

Proof of Theorem petidres2

Step	Hyp	Ref	Expression
1		disjALTVidres 39059	. 2 ⊢ Disj ( I ↾ 𝐴)
2	1	petlemi 39119	1 ⊢ (( Disj ( I ↾ 𝐴) ∧ (dom ( I ↾ 𝐴) / ( I ↾ 𝐴)) = 𝐴) ↔ ( EqvRel ≀ ( I ↾ 𝐴) ∧ (dom ≀ ( I ↾ 𝐴) / ≀ ( I ↾ 𝐴)) = 𝐴))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542   I cid 5519  dom cdm 5625   ↾ cres 5627   / cqs 8636   ≀ ccoss 38386   EqvRel weqvrel 38403   Disj wdisjALTV 38422

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3062  df-rmo 3351  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ec 8639  df-qs 8643  df-coss 38704  df-refrel 38795  df-cnvrefrel 38810  df-symrel 38827  df-trrel 38861  df-eqvrel 38872  df-funALTV 38970  df-disjALTV 38993

This theorem is referenced by:  petidres  39125