Theorem mpets 39868

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 39877, 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 39867	. . . 4 ⊢ (𝑎 ∈ V → ((◡ E ↾ 𝑎) Parts 𝑎 ↔ ≀ (◡ E ↾ 𝑎) Ers 𝑎))
2	1	elv 3456	. . 3 ⊢ ((◡ E ↾ 𝑎) Parts 𝑎 ↔ ≀ (◡ E ↾ 𝑎) Ers 𝑎)
3	2	abbii 2828	. 2 ⊢ {𝑎 ∣ (◡ E ↾ 𝑎) Parts 𝑎} = {𝑎 ∣ ≀ (◡ E ↾ 𝑎) Ers 𝑎}
4		df-membparts 39782	. 2 ⊢ MembParts = {𝑎 ∣ (◡ E ↾ 𝑎) Parts 𝑎}
5		df-comembers 39662	. 2 ⊢ CoMembErs = {𝑎 ∣ ≀ (◡ E ↾ 𝑎) Ers 𝑎}
6	3, 4, 5	3eqtr4i 2794	1 ⊢ MembParts = CoMembErs

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570  {cab 2739  Vcvv 3451   class class class wbr 5103   E cep 5550  ^◡ccnv 5650   ↾ cres 5653   ≀ ccoss 39095   Ers cers 39120   CoMembErs ccomembers 39124   Parts cparts 39135   MembParts cmembparts 39137

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

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 2100  df-mo 2565  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-id 5546  df-eprel 5551  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-ec 8712  df-qs 8716  df-rels 39352  df-coss 39413  df-coels 39414  df-ssr 39490  df-refs 39502  df-refrels 39503  df-refrel 39504  df-cnvrefs 39517  df-cnvrefrels 39518  df-cnvrefrel 39519  df-syms 39534  df-symrels 39535  df-symrel 39536  df-trs 39568  df-trrels 39569  df-trrel 39570  df-eqvrels 39580  df-eqvrel 39581  df-coeleqvrel 39583  df-dmqss 39634  df-dmqs 39635  df-ers 39660  df-erALTV 39661  df-comembers 39662  df-comember 39663  df-funALTV 39679  df-disjss 39700  df-disjs 39701  df-disjALTV 39702  df-eldisj 39704  df-parts 39780  df-part 39781  df-membparts 39782  df-membpart 39783

This theorem is used by:  petseq  39888