Theorem mpets 40154

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 40163, 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 40153	. . . 4 ⊢ (𝑎 ∈ V → ((◡ E ↾ 𝑎) Parts 𝑎 ↔ ≀ (◡ E ↾ 𝑎) Ers 𝑎))
2	1	elv 3459	. . 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 40068	. 2 ⊢ MembParts = {𝑎 ∣ (◡ E ↾ 𝑎) Parts 𝑎}
5		df-comembers 39948	. 2 ⊢ CoMembErs = {𝑎 ∣ ≀ (◡ E ↾ 𝑎) Ers 𝑎}
6	3, 4, 5	3eqtr4i 2796	1 ⊢ MembParts = CoMembErs

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570  {cab 2741  Vcvv 3454   class class class wbr 5108   E cep 5557  ^◡ccnv 5657   ↾ cres 5660   ≀ ccoss 39381   Ers cers 39406   CoMembErs ccomembers 39410   Parts cparts 39421   MembParts cmembparts 39423

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-id 5553  df-eprel 5558  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-ec 8757  df-qs 8761  df-rels 39638  df-coss 39699  df-coels 39700  df-ssr 39776  df-refs 39788  df-refrels 39789  df-refrel 39790  df-cnvrefs 39803  df-cnvrefrels 39804  df-cnvrefrel 39805  df-syms 39820  df-symrels 39821  df-symrel 39822  df-trs 39854  df-trrels 39855  df-trrel 39856  df-eqvrels 39866  df-eqvrel 39867  df-coeleqvrel 39869  df-dmqss 39920  df-dmqs 39921  df-ers 39946  df-erALTV 39947  df-comembers 39948  df-comember 39949  df-funALTV 39965  df-disjss 39986  df-disjs 39987  df-disjALTV 39988  df-eldisj 39990  df-parts 40066  df-part 40067  df-membparts 40068  df-membpart 40069

This theorem is used by:  petseq  40174