Theorem petxrnidres2 38218

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

Assertion

Ref	Expression
petxrnidres2	⊢ (( Disj (𝑅 ⋉ ( I ↾ 𝐴)) ∧ (dom (𝑅 ⋉ ( I ↾ 𝐴)) / (𝑅 ⋉ ( I ↾ 𝐴))) = 𝐴) ↔ ( EqvRel ≀ (𝑅 ⋉ ( I ↾ 𝐴)) ∧ (dom ≀ (𝑅 ⋉ ( I ↾ 𝐴)) / ≀ (𝑅 ⋉ ( I ↾ 𝐴))) = 𝐴))

Proof of Theorem petxrnidres2

Step	Hyp	Ref	Expression
1		disjALTVxrnidres 38154	. 2 ⊢ Disj (𝑅 ⋉ ( I ↾ 𝐴))
2	1	petlemi 38209	1 ⊢ (( Disj (𝑅 ⋉ ( I ↾ 𝐴)) ∧ (dom (𝑅 ⋉ ( I ↾ 𝐴)) / (𝑅 ⋉ ( I ↾ 𝐴))) = 𝐴) ↔ ( EqvRel ≀ (𝑅 ⋉ ( I ↾ 𝐴)) ∧ (dom ≀ (𝑅 ⋉ ( I ↾ 𝐴)) / ≀ (𝑅 ⋉ ( I ↾ 𝐴))) = 𝐴))

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1534   I cid 5569  dom cdm 5672   ↾ cres 5674   / cqs 8715   ⋉ cxrn 37569   ≀ ccoss 37570   EqvRel weqvrel 37587   Disj wdisjALTV 37604

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7732

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rmo 3371  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fo 6548  df-fv 6550  df-1st 7985  df-2nd 7986  df-ec 8718  df-qs 8722  df-xrn 37767  df-coss 37807  df-refrel 37908  df-cnvrefrel 37923  df-symrel 37940  df-trrel 37970  df-eqvrel 37981  df-funALTV 38078  df-disjALTV 38101

This theorem is referenced by:  petxrnidres  38219