Theorem mpet3 37050

Description: Member Partition-Equivalence Theorem. Together with mpet 37053 mpet2 37054, mostly in its conventional cpet 37052 and cpet2 37051 form, this is what we used to think of as the partition equivalence theorem (but cf. pet2 37064 with general 𝑅). (Contributed by Peter Mazsa, 4-May-2018.) (Revised by Peter Mazsa, 26-Sep-2021.)

Assertion

Ref	Expression
mpet3	⊢ (( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴) ↔ ( CoElEqvRel 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴))

Proof of Theorem mpet3

Step	Hyp	Ref	Expression
1		eldisjn0elb 36959	. 2 ⊢ (( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴) ↔ ( Disj (◡ E ↾ 𝐴) ∧ (dom (◡ E ↾ 𝐴) / (◡ E ↾ 𝐴)) = 𝐴))
2		eqvrelqseqdisj3 37045	. . 3 ⊢ (( EqvRel ≀ (◡ E ↾ 𝐴) ∧ (dom ≀ (◡ E ↾ 𝐴) / ≀ (◡ E ↾ 𝐴)) = 𝐴) → Disj (◡ E ↾ 𝐴))
3	2	petlem 37026	. 2 ⊢ (( Disj (◡ E ↾ 𝐴) ∧ (dom (◡ E ↾ 𝐴) / (◡ E ↾ 𝐴)) = 𝐴) ↔ ( EqvRel ≀ (◡ E ↾ 𝐴) ∧ (dom ≀ (◡ E ↾ 𝐴) / ≀ (◡ E ↾ 𝐴)) = 𝐴))
4		eqvreldmqs 36889	. 2 ⊢ (( EqvRel ≀ (◡ E ↾ 𝐴) ∧ (dom ≀ (◡ E ↾ 𝐴) / ≀ (◡ E ↾ 𝐴)) = 𝐴) ↔ ( CoElEqvRel 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴))
5	1, 3, 4	3bitri 297	1 ⊢ (( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴) ↔ ( CoElEqvRel 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴))

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 397   = wceq 1539   ∈ wcel 2104  ∅c0 4262  ∪ cuni 4844   E cep 5505  ^◡ccnv 5599  dom cdm 5600   ↾ cres 5602   / cqs 8528   ≀ ccoss 36381   ∼ ccoels 36382   EqvRel weqvrel 36398   CoElEqvRel wcoeleqvrel 36400   Disj wdisjALTV 36415   ElDisj weldisj 36417

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pr 5361

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3331  df-rab 3333  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-br 5082  df-opab 5144  df-id 5500  df-eprel 5506  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-ec 8531  df-qs 8535  df-coss 36625  df-coels 36626  df-refrel 36726  df-cnvrefrel 36741  df-symrel 36758  df-trrel 36788  df-eqvrel 36799  df-coeleqvrel 36801  df-funALTV 36896  df-disjALTV 36919  df-eldisj 36921

This theorem is referenced by:  mpet  37053