Theorem petxrnidres 38779

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 38407, disjALTVxrnidres 38714 and eqvrel1cossxrnidres 38748. (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 38778	. 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 38725	. 2 ⊢ ((𝑅 ⋉ ( I ↾ 𝐴)) Part 𝐴 ↔ ( Disj (𝑅 ⋉ ( I ↾ 𝐴)) ∧ (dom (𝑅 ⋉ ( I ↾ 𝐴)) / (𝑅 ⋉ ( I ↾ 𝐴))) = 𝐴))
3		dferALTV2 38624	. 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 1537   I cid 5592  dom cdm 5700   ↾ cres 5702   / cqs 8762   ⋉ cxrn 38134   ≀ ccoss 38135   EqvRel weqvrel 38152   ErALTV werALTV 38161   Disj wdisjALTV 38169   Part wpart 38174

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fo 6579  df-fv 6581  df-1st 8030  df-2nd 8031  df-ec 8765  df-qs 8769  df-xrn 38327  df-coss 38367  df-refrel 38468  df-cnvrefrel 38483  df-symrel 38500  df-trrel 38530  df-eqvrel 38541  df-dmqs 38595  df-erALTV 38620  df-funALTV 38638  df-disjALTV 38661  df-part 38722

This theorem is referenced by: (None)