Theorem mpet3 39461

Description: Member Partition-Equivalence Theorem. Together with mpet 39464 mpet2 39465, mostly in its conventional cpet 39463 and cpet2 39462 form, this is what we used to think of as the partition equivalence theorem (but cf. pet2 39475 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 39356	. 2 ⊢ (( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴) ↔ ( Disj (◡ E ↾ 𝐴) ∧ (dom (◡ E ↾ 𝐴) / (◡ E ↾ 𝐴)) = 𝐴))
2		eqvrelqseqdisj3 39456	. . 3 ⊢ (( EqvRel ≀ (◡ E ↾ 𝐴) ∧ (dom ≀ (◡ E ↾ 𝐴) / ≀ (◡ E ↾ 𝐴)) = 𝐴) → Disj (◡ E ↾ 𝐴))
3	2	petlem 39426	. 2 ⊢ (( Disj (◡ E ↾ 𝐴) ∧ (dom (◡ E ↾ 𝐴) / (◡ E ↾ 𝐴)) = 𝐴) ↔ ( EqvRel ≀ (◡ E ↾ 𝐴) ∧ (dom ≀ (◡ E ↾ 𝐴) / ≀ (◡ E ↾ 𝐴)) = 𝐴))
4		eqvreldmqs 39271	. 2 ⊢ (( EqvRel ≀ (◡ E ↾ 𝐴) ∧ (dom ≀ (◡ E ↾ 𝐴) / ≀ (◡ E ↾ 𝐴)) = 𝐴) ↔ ( CoElEqvRel 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴))
5	1, 3, 4	3bitri 300	1 ⊢ (( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴) ↔ ( CoElEqvRel 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴))

Syntax hints:  ¬ wn 3   ↔ wb 209   ∧ wa 400   = wceq 1563   ∈ wcel 2145  ∅c0 4288  ∪ cuni 4868   E cep 5551  ^◡ccnv 5651  dom cdm 5652   ↾ cres 5654   / cqs 8681   ≀ ccoss 38694   ∼ ccoels 38695   EqvRel weqvrel 38711   CoElEqvRel wcoeleqvrel 38713   Disj wdisjALTV 38730   ElDisj weldisj 38732

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rmo 3370  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-id 5547  df-eprel 5552  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-ec 8684  df-qs 8688  df-coss 39012  df-coels 39013  df-refrel 39103  df-cnvrefrel 39118  df-symrel 39135  df-trrel 39169  df-eqvrel 39180  df-coeleqvrel 39182  df-funALTV 39278  df-disjALTV 39301  df-eldisj 39303

This theorem is referenced by:  mpet  39464