Theorem mpets 39997

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 40006, 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 39996	. . . 4 ⊢ (𝑎 ∈ V → ((◡ E ↾ 𝑎) Parts 𝑎 ↔ ≀ (◡ E ↾ 𝑎) Ers 𝑎))
2	1	elv 3455	. . 3 ⊢ ((◡ E ↾ 𝑎) Parts 𝑎 ↔ ≀ (◡ E ↾ 𝑎) Ers 𝑎)
3	2	abbii 2827	. 2 ⊢ {𝑎 ∣ (◡ E ↾ 𝑎) Parts 𝑎} = {𝑎 ∣ ≀ (◡ E ↾ 𝑎) Ers 𝑎}
4		df-membparts 39911	. 2 ⊢ MembParts = {𝑎 ∣ (◡ E ↾ 𝑎) Parts 𝑎}
5		df-comembers 39791	. 2 ⊢ CoMembErs = {𝑎 ∣ ≀ (◡ E ↾ 𝑎) Ers 𝑎}
6	3, 4, 5	3eqtr4i 2793	1 ⊢ MembParts = CoMembErs

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1569  {cab 2738  Vcvv 3450   class class class wbr 5102   E cep 5550  ^◡ccnv 5650   ↾ cres 5653   ≀ ccoss 39224   Ers cers 39249   CoMembErs ccomembers 39253   Parts cparts 39264   MembParts cmembparts 39266

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  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 8726  df-qs 8730  df-rels 39481  df-coss 39542  df-coels 39543  df-ssr 39619  df-refs 39631  df-refrels 39632  df-refrel 39633  df-cnvrefs 39646  df-cnvrefrels 39647  df-cnvrefrel 39648  df-syms 39663  df-symrels 39664  df-symrel 39665  df-trs 39697  df-trrels 39698  df-trrel 39699  df-eqvrels 39709  df-eqvrel 39710  df-coeleqvrel 39712  df-dmqss 39763  df-dmqs 39764  df-ers 39789  df-erALTV 39790  df-comembers 39791  df-comember 39792  df-funALTV 39808  df-disjss 39829  df-disjs 39830  df-disjALTV 39831  df-eldisj 39833  df-parts 39909  df-part 39910  df-membparts 39911  df-membpart 39912

This theorem is used by:  petseq  40017