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Theorem mpets 38314
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 38323, 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 38313 . . . 4 (𝑎 ∈ V → (( E ↾ 𝑎) Parts 𝑎 ↔ ≀ ( E ↾ 𝑎) Ers 𝑎))
21elv 3477 . . 3 (( E ↾ 𝑎) Parts 𝑎 ↔ ≀ ( E ↾ 𝑎) Ers 𝑎)
32abbii 2798 . 2 {𝑎 ∣ ( E ↾ 𝑎) Parts 𝑎} = {𝑎 ∣ ≀ ( E ↾ 𝑎) Ers 𝑎}
4 df-membparts 38239 . 2 MembParts = {𝑎 ∣ ( E ↾ 𝑎) Parts 𝑎}
5 df-comembers 38137 . 2 CoMembErs = {𝑎 ∣ ≀ ( E ↾ 𝑎) Ers 𝑎}
63, 4, 53eqtr4i 2766 1 MembParts = CoMembErs
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1534  {cab 2705  Vcvv 3471   class class class wbr 5148   E cep 5581  ccnv 5677  cres 5680  ccoss 37648   Ers cers 37673   CoMembErs ccomembers 37675   Parts cparts 37686   MembParts cmembparts 37688
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3373  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-id 5576  df-eprel 5582  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ec 8727  df-qs 8731  df-coss 37883  df-coels 37884  df-rels 37957  df-ssr 37970  df-refs 37982  df-refrels 37983  df-refrel 37984  df-cnvrefs 37997  df-cnvrefrels 37998  df-cnvrefrel 37999  df-syms 38014  df-symrels 38015  df-symrel 38016  df-trs 38044  df-trrels 38045  df-trrel 38046  df-eqvrels 38056  df-eqvrel 38057  df-coeleqvrel 38059  df-dmqss 38110  df-dmqs 38111  df-ers 38135  df-erALTV 38136  df-comembers 38137  df-comember 38138  df-funALTV 38154  df-disjss 38175  df-disjs 38176  df-disjALTV 38177  df-eldisj 38179  df-parts 38237  df-part 38238  df-membparts 38239  df-membpart 38240
This theorem is referenced by: (None)
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