Theorem mpets 39323

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 39332, 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 39322	. . . 4 ⊢ (𝑎 ∈ V → ((◡ E ↾ 𝑎) Parts 𝑎 ↔ ≀ (◡ E ↾ 𝑎) Ers 𝑎))
2	1	elv 3436	. . 3 ⊢ ((◡ E ↾ 𝑎) Parts 𝑎 ↔ ≀ (◡ E ↾ 𝑎) Ers 𝑎)
3	2	abbii 2806	. 2 ⊢ {𝑎 ∣ (◡ E ↾ 𝑎) Parts 𝑎} = {𝑎 ∣ ≀ (◡ E ↾ 𝑎) Ers 𝑎}
4		df-membparts 39237	. 2 ⊢ MembParts = {𝑎 ∣ (◡ E ↾ 𝑎) Parts 𝑎}
5		df-comembers 39117	. 2 ⊢ CoMembErs = {𝑎 ∣ ≀ (◡ E ↾ 𝑎) Ers 𝑎}
6	3, 4, 5	3eqtr4i 2772	1 ⊢ MembParts = CoMembErs

Syntax hints:   ↔ wb 207   = wceq 1547  {cab 2717  Vcvv 3431   class class class wbr 5072   E cep 5517  ^◡ccnv 5617   ↾ cres 5620   ≀ ccoss 38550   Ers cers 38575   CoMembErs ccomembers 38579   Parts cparts 38590   MembParts cmembparts 38592

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rmo 3344  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-id 5513  df-eprel 5518  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-ec 8635  df-qs 8639  df-rels 38807  df-coss 38868  df-coels 38869  df-ssr 38945  df-refs 38957  df-refrels 38958  df-refrel 38959  df-cnvrefs 38972  df-cnvrefrels 38973  df-cnvrefrel 38974  df-syms 38989  df-symrels 38990  df-symrel 38991  df-trs 39023  df-trrels 39024  df-trrel 39025  df-eqvrels 39035  df-eqvrel 39036  df-coeleqvrel 39038  df-dmqss 39089  df-dmqs 39090  df-ers 39115  df-erALTV 39116  df-comembers 39117  df-comember 39118  df-funALTV 39134  df-disjss 39155  df-disjs 39156  df-disjALTV 39157  df-eldisj 39159  df-parts 39235  df-part 39236  df-membparts 39237  df-membpart 39238

This theorem is referenced by:  petseq  39343