Theorem pet2 39427

Description: Partition-Equivalence Theorem, with general 𝑅. This theorem (together with pet 39428 and pets 39429) is the main result of my investigation into set theory, see the comment of pet 39428. (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 39410	. 2 ⊢ (( EqvRel ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ (dom ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) / ≀ (𝑅 ⋉ (◡ E ↾ 𝐴))) = 𝐴) → Disj (𝑅 ⋉ (◡ E ↾ 𝐴)))
2	1	petlem 39378	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 208   ∧ wa 399   = wceq 1559   E cep 5544  ^◡ccnv 5644  dom cdm 5645   ↾ cres 5647   / cqs 8672   ⋉ cxrn 38637   ≀ ccoss 38646   EqvRel weqvrel 38663   Disj wdisjALTV 38682

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-eprel 5545  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-fo 6523  df-fv 6525  df-1st 7966  df-2nd 7967  df-ec 8675  df-qs 8679  df-xrn 38843  df-coss 38964  df-refrel 39055  df-cnvrefrel 39070  df-symrel 39087  df-trrel 39121  df-eqvrel 39132  df-funALTV 39230  df-disjALTV 39253  df-eldisj 39255

This theorem is referenced by:  pet  39428