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Theorem mpets 38216
Description: Member Partition-Equivalence Theorem in its shortest possible form: it shows that member partitions and comember equivalence relations are literally the same. Cf. pet 38225, the Partition-Equivalence Theorem, with general 𝑅. (Contributed by Peter Mazsa, 31-Dec-2024.)
Assertion
Ref Expression
mpets MembParts = CoMembErs

Proof of Theorem mpets
StepHypRef Expression
1 mpets2 38215 . . . 4 (𝑎 ∈ V → (( E ↾ 𝑎) Parts 𝑎 ↔ ≀ ( E ↾ 𝑎) Ers 𝑎))
21elv 3472 . . 3 (( E ↾ 𝑎) Parts 𝑎 ↔ ≀ ( E ↾ 𝑎) Ers 𝑎)
32abbii 2794 . 2 {𝑎 ∣ ( E ↾ 𝑎) Parts 𝑎} = {𝑎 ∣ ≀ ( E ↾ 𝑎) Ers 𝑎}
4 df-membparts 38141 . 2 MembParts = {𝑎 ∣ ( E ↾ 𝑎) Parts 𝑎}
5 df-comembers 38039 . 2 CoMembErs = {𝑎 ∣ ≀ ( E ↾ 𝑎) Ers 𝑎}
63, 4, 53eqtr4i 2762 1 MembParts = CoMembErs
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1533  {cab 2701  Vcvv 3466   class class class wbr 5139   E cep 5570  ccnv 5666  cres 5669  ccoss 37547   Ers cers 37572   CoMembErs ccomembers 37574   Parts cparts 37585   MembParts cmembparts 37587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-id 5565  df-eprel 5571  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-ec 8702  df-qs 8706  df-coss 37785  df-coels 37786  df-rels 37859  df-ssr 37872  df-refs 37884  df-refrels 37885  df-refrel 37886  df-cnvrefs 37899  df-cnvrefrels 37900  df-cnvrefrel 37901  df-syms 37916  df-symrels 37917  df-symrel 37918  df-trs 37946  df-trrels 37947  df-trrel 37948  df-eqvrels 37958  df-eqvrel 37959  df-coeleqvrel 37961  df-dmqss 38012  df-dmqs 38013  df-ers 38037  df-erALTV 38038  df-comembers 38039  df-comember 38040  df-funALTV 38056  df-disjss 38077  df-disjs 38078  df-disjALTV 38079  df-eldisj 38081  df-parts 38139  df-part 38140  df-membparts 38141  df-membpart 38142
This theorem is referenced by: (None)
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