Theorem mpets 39605

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 39614, 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 39604	. . . 4 ⊢ (𝑎 ∈ V → ((◡ E ↾ 𝑎) Parts 𝑎 ↔ ≀ (◡ E ↾ 𝑎) Ers 𝑎))
2	1	elv 3460	. . 3 ⊢ ((◡ E ↾ 𝑎) Parts 𝑎 ↔ ≀ (◡ E ↾ 𝑎) Ers 𝑎)
3	2	abbii 2830	. 2 ⊢ {𝑎 ∣ (◡ E ↾ 𝑎) Parts 𝑎} = {𝑎 ∣ ≀ (◡ E ↾ 𝑎) Ers 𝑎}
4		df-membparts 39519	. 2 ⊢ MembParts = {𝑎 ∣ (◡ E ↾ 𝑎) Parts 𝑎}
5		df-comembers 39399	. 2 ⊢ CoMembErs = {𝑎 ∣ ≀ (◡ E ↾ 𝑎) Ers 𝑎}
6	3, 4, 5	3eqtr4i 2796	1 ⊢ MembParts = CoMembErs

Syntax hints:   ↔ wb 209   = wceq 1570  {cab 2741  Vcvv 3455   class class class wbr 5109   E cep 5560  ^◡ccnv 5660   ↾ cres 5663   ≀ ccoss 38832   Ers cers 38857   CoMembErs ccomembers 38861   Parts cparts 38872   MembParts cmembparts 38874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-id 5556  df-eprel 5561  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-ec 8692  df-qs 8696  df-rels 39089  df-coss 39150  df-coels 39151  df-ssr 39227  df-refs 39239  df-refrels 39240  df-refrel 39241  df-cnvrefs 39254  df-cnvrefrels 39255  df-cnvrefrel 39256  df-syms 39271  df-symrels 39272  df-symrel 39273  df-trs 39305  df-trrels 39306  df-trrel 39307  df-eqvrels 39317  df-eqvrel 39318  df-coeleqvrel 39320  df-dmqss 39371  df-dmqs 39372  df-ers 39397  df-erALTV 39398  df-comembers 39399  df-comember 39400  df-funALTV 39416  df-disjss 39437  df-disjs 39438  df-disjALTV 39439  df-eldisj 39441  df-parts 39517  df-part 39518  df-membparts 39519  df-membpart 39520

This theorem is referenced by:  petseq  39625