Theorem petxrnidres 38860

Description: A class 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. Cf. br1cossxrnidres 38487, disjALTVxrnidres 38795 and eqvrel1cossxrnidres 38829. (Contributed by Peter Mazsa, 31-Dec-2021.)

Assertion

Ref	Expression
petxrnidres	⊢ ((𝑅 ⋉ ( I ↾ 𝐴)) Part 𝐴 ↔ ≀ (𝑅 ⋉ ( I ↾ 𝐴)) ErALTV 𝐴)

Proof of Theorem petxrnidres

Step	Hyp	Ref	Expression
1		petxrnidres2 38859	. 2 ⊢ (( Disj (𝑅 ⋉ ( I ↾ 𝐴)) ∧ (dom (𝑅 ⋉ ( I ↾ 𝐴)) / (𝑅 ⋉ ( I ↾ 𝐴))) = 𝐴) ↔ ( EqvRel ≀ (𝑅 ⋉ ( I ↾ 𝐴)) ∧ (dom ≀ (𝑅 ⋉ ( I ↾ 𝐴)) / ≀ (𝑅 ⋉ ( I ↾ 𝐴))) = 𝐴))
2		dfpart2 38806	. 2 ⊢ ((𝑅 ⋉ ( I ↾ 𝐴)) Part 𝐴 ↔ ( Disj (𝑅 ⋉ ( I ↾ 𝐴)) ∧ (dom (𝑅 ⋉ ( I ↾ 𝐴)) / (𝑅 ⋉ ( I ↾ 𝐴))) = 𝐴))
3		dferALTV2 38705	. 2 ⊢ ( ≀ (𝑅 ⋉ ( I ↾ 𝐴)) ErALTV 𝐴 ↔ ( EqvRel ≀ (𝑅 ⋉ ( I ↾ 𝐴)) ∧ (dom ≀ (𝑅 ⋉ ( I ↾ 𝐴)) / ≀ (𝑅 ⋉ ( I ↾ 𝐴))) = 𝐴))
4	1, 2, 3	3bitr4i 303	1 ⊢ ((𝑅 ⋉ ( I ↾ 𝐴)) Part 𝐴 ↔ ≀ (𝑅 ⋉ ( I ↾ 𝐴)) ErALTV 𝐴)

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541   I cid 5510  dom cdm 5616   ↾ cres 5618   / cqs 8621   ⋉ cxrn 38213   ≀ ccoss 38214   EqvRel weqvrel 38231   ErALTV werALTV 38240   Disj wdisjALTV 38248   Part wpart 38253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-fo 6487  df-fv 6489  df-1st 7921  df-2nd 7922  df-ec 8624  df-qs 8628  df-xrn 38398  df-coss 38447  df-refrel 38548  df-cnvrefrel 38563  df-symrel 38580  df-trrel 38610  df-eqvrel 38621  df-dmqs 38675  df-erALTV 38701  df-funALTV 38719  df-disjALTV 38742  df-part 38803

This theorem is referenced by: (None)