Theorem petxrnidres2 39994

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 39927	. 2 ⊢ Disj (𝑅 ⋉ ( I ↾ 𝐴))
2	1	petlemi 39985	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))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1569   I cid 5545  dom cdm 5651   ↾ cres 5653   / cqs 8724   ⋉ cxrn 39243   ≀ ccoss 39252   EqvRel weqvrel 39269   Disj wdisjALTV 39288

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  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-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-fv 6551  df-1st 8005  df-2nd 8006  df-ec 8727  df-qs 8731  df-xrn 39449  df-coss 39570  df-refrel 39661  df-cnvrefrel 39676  df-symrel 39693  df-trrel 39727  df-eqvrel 39738  df-funALTV 39836  df-disjALTV 39859

This theorem is used by:  petxrnidres  39995