Theorem petidres 38323

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 38294. (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 38322	. 2 ⊢ (( Disj ( I ↾ 𝐴) ∧ (dom ( I ↾ 𝐴) / ( I ↾ 𝐴)) = 𝐴) ↔ ( EqvRel ≀ ( I ↾ 𝐴) ∧ (dom ≀ ( I ↾ 𝐴) / ≀ ( I ↾ 𝐴)) = 𝐴))
2		dfpart2 38273	. 2 ⊢ (( I ↾ 𝐴) Part 𝐴 ↔ ( Disj ( I ↾ 𝐴) ∧ (dom ( I ↾ 𝐴) / ( I ↾ 𝐴)) = 𝐴))
3		dferALTV2 38172	. 2 ⊢ ( ≀ ( I ↾ 𝐴) ErALTV 𝐴 ↔ ( EqvRel ≀ ( I ↾ 𝐴) ∧ (dom ≀ ( I ↾ 𝐴) / ≀ ( I ↾ 𝐴)) = 𝐴))
4	1, 2, 3	3bitr4i 302	1 ⊢ (( I ↾ 𝐴) Part 𝐴 ↔ ≀ ( I ↾ 𝐴) ErALTV 𝐴)

Syntax hints:   ↔ wb 205   ∧ wa 394   = wceq 1533   I cid 5579  dom cdm 5682   ↾ cres 5684   / cqs 8730   ≀ ccoss 37681   EqvRel weqvrel 37698   ErALTV werALTV 37707   Disj wdisjALTV 37715   Part wpart 37720

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rmo 3374  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-ec 8733  df-qs 8737  df-coss 37915  df-refrel 38016  df-cnvrefrel 38031  df-symrel 38048  df-trrel 38078  df-eqvrel 38089  df-dmqs 38143  df-erALTV 38168  df-funALTV 38186  df-disjALTV 38209  df-part 38270

This theorem is referenced by: (None)