Theorem mpets 39452

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 39461, 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 39451	. . . 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 2829	. 2 ⊢ {𝑎 ∣ (◡ E ↾ 𝑎) Parts 𝑎} = {𝑎 ∣ ≀ (◡ E ↾ 𝑎) Ers 𝑎}
4		df-membparts 39366	. 2 ⊢ MembParts = {𝑎 ∣ (◡ E ↾ 𝑎) Parts 𝑎}
5		df-comembers 39246	. 2 ⊢ CoMembErs = {𝑎 ∣ ≀ (◡ E ↾ 𝑎) Ers 𝑎}
6	3, 4, 5	3eqtr4i 2795	1 ⊢ MembParts = CoMembErs

Syntax hints:   ↔ wb 208   = wceq 1560  {cab 2740  Vcvv 3454   class class class wbr 5100   E cep 5546  ^◡ccnv 5646   ↾ cres 5649   ≀ ccoss 38679   Ers cers 38704   CoMembErs ccomembers 38708   Parts cparts 38719   MembParts cmembparts 38721

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rmo 3367  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-id 5542  df-eprel 5547  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-ec 8680  df-qs 8684  df-rels 38936  df-coss 38997  df-coels 38998  df-ssr 39074  df-refs 39086  df-refrels 39087  df-refrel 39088  df-cnvrefs 39101  df-cnvrefrels 39102  df-cnvrefrel 39103  df-syms 39118  df-symrels 39119  df-symrel 39120  df-trs 39152  df-trrels 39153  df-trrel 39154  df-eqvrels 39164  df-eqvrel 39165  df-coeleqvrel 39167  df-dmqss 39218  df-dmqs 39219  df-ers 39244  df-erALTV 39245  df-comembers 39246  df-comember 39247  df-funALTV 39263  df-disjss 39284  df-disjs 39285  df-disjALTV 39286  df-eldisj 39288  df-parts 39364  df-part 39365  df-membparts 39366  df-membpart 39367

This theorem is referenced by:  petseq  39472