Theorem petidres 39591

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 39562. (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 39590	. 2 ⊢ (( Disj ( I ↾ 𝐴) ∧ (dom ( I ↾ 𝐴) / ( I ↾ 𝐴)) = 𝐴) ↔ ( EqvRel ≀ ( I ↾ 𝐴) ∧ (dom ≀ ( I ↾ 𝐴) / ≀ ( I ↾ 𝐴)) = 𝐴))
2		dfpart2 39541	. 2 ⊢ (( I ↾ 𝐴) Part 𝐴 ↔ ( Disj ( I ↾ 𝐴) ∧ (dom ( I ↾ 𝐴) / ( I ↾ 𝐴)) = 𝐴))
3		dferALTV2 39422	. 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)

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1570   I cid 5555  dom cdm 5661   ↾ cres 5663   / cqs 8689   ≀ ccoss 38852   EqvRel weqvrel 38869   ErALTV werALTV 38878   Disj wdisjALTV 38888   Part wpart 38893

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080  df-rex 3090  df-rmo 3369  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-ec 8692  df-qs 8696  df-coss 39170  df-refrel 39261  df-cnvrefrel 39276  df-symrel 39293  df-trrel 39327  df-eqvrel 39338  df-dmqs 39392  df-erALTV 39418  df-funALTV 39436  df-disjALTV 39459  df-part 39538

This theorem is referenced by: (None)