Theorem mpets 39494

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 39503, the Partition-Equivalence Theorem, with general 𝑅. (Contributed by Peter Mazsa, 31-Dec-2024.)

Assertion

Ref	Expression
mpets	⊢ MembParts = CoMembErs

Proof of Theorem mpets

Step	Hyp	Ref	Expression
1		mpets2 39493	. . . 4 ⊢ (𝑎 ∈ V → ((◡ E ↾ 𝑎) Parts 𝑎 ↔ ≀ (◡ E ↾ 𝑎) Ers 𝑎))
2	1	elv 3468	. . 3 ⊢ ((◡ E ↾ 𝑎) Parts 𝑎 ↔ ≀ (◡ E ↾ 𝑎) Ers 𝑎)
3	2	abbii 2836	. 2 ⊢ {𝑎 ∣ (◡ E ↾ 𝑎) Parts 𝑎} = {𝑎 ∣ ≀ (◡ E ↾ 𝑎) Ers 𝑎}
4		df-membparts 39408	. 2 ⊢ MembParts = {𝑎 ∣ (◡ E ↾ 𝑎) Parts 𝑎}
5		df-comembers 39288	. 2 ⊢ CoMembErs = {𝑎 ∣ ≀ (◡ E ↾ 𝑎) Ers 𝑎}
6	3, 4, 5	3eqtr4i 2802	1 ⊢ MembParts = CoMembErs

Syntax hints:   ↔ wb 209   = wceq 1567  {cab 2747  Vcvv 3463   class class class wbr 5113   E cep 5561  ^◡ccnv 5661   ↾ cres 5664   ≀ ccoss 38721   Ers cers 38746   CoMembErs ccomembers 38750   Parts cparts 38761   MembParts cmembparts 38763

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-id 5557  df-eprel 5562  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-ec 8695  df-qs 8699  df-rels 38978  df-coss 39039  df-coels 39040  df-ssr 39116  df-refs 39128  df-refrels 39129  df-refrel 39130  df-cnvrefs 39143  df-cnvrefrels 39144  df-cnvrefrel 39145  df-syms 39160  df-symrels 39161  df-symrel 39162  df-trs 39194  df-trrels 39195  df-trrel 39196  df-eqvrels 39206  df-eqvrel 39207  df-coeleqvrel 39209  df-dmqss 39260  df-dmqs 39261  df-ers 39286  df-erALTV 39287  df-comembers 39288  df-comember 39289  df-funALTV 39305  df-disjss 39326  df-disjs 39327  df-disjALTV 39328  df-eldisj 39330  df-parts 39406  df-part 39407  df-membparts 39408  df-membpart 39409

This theorem is referenced by:  petseq  39514