Theorem pet2 39299

Description: Partition-Equivalence Theorem, with general 𝑅. This theorem (together with pet 39300 and pets 39301) is the main result of my investigation into set theory, see the comment of pet 39300. (Contributed by Peter Mazsa, 24-May-2021.) (Revised by Peter Mazsa, 23-Sep-2021.)

Assertion

Ref	Expression
pet2	⊢ (( Disj (𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ (dom (𝑅 ⋉ (◡ E ↾ 𝐴)) / (𝑅 ⋉ (◡ E ↾ 𝐴))) = 𝐴) ↔ ( EqvRel ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ (dom ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) / ≀ (𝑅 ⋉ (◡ E ↾ 𝐴))) = 𝐴))

Proof of Theorem pet2

Step	Hyp	Ref	Expression
1		eqvrelqseqdisj5 39282	. 2 ⊢ (( EqvRel ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ (dom ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) / ≀ (𝑅 ⋉ (◡ E ↾ 𝐴))) = 𝐴) → Disj (𝑅 ⋉ (◡ E ↾ 𝐴)))
2	1	petlem 39250	1 ⊢ (( Disj (𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ (dom (𝑅 ⋉ (◡ E ↾ 𝐴)) / (𝑅 ⋉ (◡ E ↾ 𝐴))) = 𝐴) ↔ ( EqvRel ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ (dom ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) / ≀ (𝑅 ⋉ (◡ E ↾ 𝐴))) = 𝐴))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542   E cep 5523  ^◡ccnv 5623  dom cdm 5624   ↾ cres 5626   / cqs 8635   ⋉ cxrn 38509   ≀ ccoss 38518   EqvRel weqvrel 38535   Disj wdisjALTV 38554

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-eprel 5524  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fo 6498  df-fv 6500  df-1st 7935  df-2nd 7936  df-ec 8638  df-qs 8642  df-xrn 38715  df-coss 38836  df-refrel 38927  df-cnvrefrel 38942  df-symrel 38959  df-trrel 38993  df-eqvrel 39004  df-funALTV 39102  df-disjALTV 39125  df-eldisj 39127

This theorem is referenced by:  pet  39300