Theorem petidres 39854

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 39825. (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 39853	. 2 ⊢ (( Disj ( I ↾ 𝐴) ∧ (dom ( I ↾ 𝐴) / ( I ↾ 𝐴)) = 𝐴) ↔ ( EqvRel ≀ ( I ↾ 𝐴) ∧ (dom ≀ ( I ↾ 𝐴) / ≀ ( I ↾ 𝐴)) = 𝐴))
2		dfpart2 39804	. 2 ⊢ (( I ↾ 𝐴) Part 𝐴 ↔ ( Disj ( I ↾ 𝐴) ∧ (dom ( I ↾ 𝐴) / ( I ↾ 𝐴)) = 𝐴))
3		dferALTV2 39685	. 2 ⊢ ( ≀ ( I ↾ 𝐴) ErALTV 𝐴 ↔ ( EqvRel ≀ ( I ↾ 𝐴) ∧ (dom ≀ ( I ↾ 𝐴) / ≀ ( I ↾ 𝐴)) = 𝐴))
4	1, 2, 3	3bitr4i 306	1 ⊢ (( I ↾ 𝐴) Part 𝐴 ↔ ≀ ( I ↾ 𝐴) ErALTV 𝐴)

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   I cid 5545  dom cdm 5651   ↾ cres 5653   / cqs 8716   ≀ ccoss 39115   EqvRel weqvrel 39132   ErALTV werALTV 39141   Disj wdisjALTV 39151   Part wpart 39156

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-pr 5391

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 2100  df-mo 2565  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3078  df-rex 3088  df-rmo 3366  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-ec 8719  df-qs 8723  df-coss 39433  df-refrel 39524  df-cnvrefrel 39539  df-symrel 39556  df-trrel 39590  df-eqvrel 39601  df-dmqs 39655  df-erALTV 39681  df-funALTV 39699  df-disjALTV 39722  df-part 39801

This theorem is used by: (None)