Theorem petseq 40099

Description: Generalized partition-equivalence identification.

The partition-side scheme PetParts and the equivalence-side scheme PetErs define the same class of spans (pairs ⟨𝑟, 𝑛⟩).

This plays the same organizational role for lifted spans that mpets 40079 plays for carriers: mpets 40079 identifies MembParts with CoMembErs at the membership-carrier level, while petseq 40099 identifies the corresponding span-level predicates built from Parts and Ers.

Unlike the earlier broad pets 40089, the bridge used here is the type-safe span theorem typesafepets 40098, which restricts to membership block-carriers. Since typedness (𝑟 ∈ Rels and the appropriate carrier condition) is now built directly into PetParts and PetErs, this theorem can be used downstream without repeatedly re-establishing basic typing premises. (Contributed by Peter Mazsa, 19-Feb-2026.)

Assertion

Ref	Expression
petseq	⊢ PetParts = PetErs

Proof of Theorem petseq

Dummy variables 𝑛 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		typesafepets 40098	. . . . . . 7 ⊢ ((𝑛 ∈ MembParts ∧ 𝑟 ∈ V) → ((𝑟 ⋉ (◡ E ↾ 𝑛)) Parts 𝑛 ↔ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛))
2	1	elvd 3460	. . . . . 6 ⊢ (𝑛 ∈ MembParts → ((𝑟 ⋉ (◡ E ↾ 𝑛)) Parts 𝑛 ↔ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛))
3	2	adantl 487	. . . . 5 ⊢ ((𝑟 ∈ Rels ∧ 𝑛 ∈ MembParts ) → ((𝑟 ⋉ (◡ E ↾ 𝑛)) Parts 𝑛 ↔ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛))
4	3	pm5.32i 585	. . . 4 ⊢ (((𝑟 ∈ Rels ∧ 𝑛 ∈ MembParts ) ∧ (𝑟 ⋉ (◡ E ↾ 𝑛)) Parts 𝑛) ↔ ((𝑟 ∈ Rels ∧ 𝑛 ∈ MembParts ) ∧ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛))
5		mpets 40079	. . . . . 6 ⊢ MembParts = CoMembErs
6	5	eleq2i 2856	. . . . 5 ⊢ (𝑛 ∈ MembParts ↔ 𝑛 ∈ CoMembErs )
7	6	anbi2i 635	. . . 4 ⊢ ((𝑟 ∈ Rels ∧ 𝑛 ∈ MembParts ) ↔ (𝑟 ∈ Rels ∧ 𝑛 ∈ CoMembErs ))
8	4, 7	bianbi 639	. . 3 ⊢ (((𝑟 ∈ Rels ∧ 𝑛 ∈ MembParts ) ∧ (𝑟 ⋉ (◡ E ↾ 𝑛)) Parts 𝑛) ↔ ((𝑟 ∈ Rels ∧ 𝑛 ∈ CoMembErs ) ∧ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛))
9	8	opabbii 5175	. 2 ⊢ {⟨𝑟, 𝑛⟩ ∣ ((𝑟 ∈ Rels ∧ 𝑛 ∈ MembParts ) ∧ (𝑟 ⋉ (◡ E ↾ 𝑛)) Parts 𝑛)} = {⟨𝑟, 𝑛⟩ ∣ ((𝑟 ∈ Rels ∧ 𝑛 ∈ CoMembErs ) ∧ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛)}
10		df-petparts 40091	. 2 ⊢ PetParts = {⟨𝑟, 𝑛⟩ ∣ ((𝑟 ∈ Rels ∧ 𝑛 ∈ MembParts ) ∧ (𝑟 ⋉ (◡ E ↾ 𝑛)) Parts 𝑛)}
11		df-peters 40092	. 2 ⊢ PetErs = {⟨𝑟, 𝑛⟩ ∣ ((𝑟 ∈ Rels ∧ 𝑛 ∈ CoMembErs ) ∧ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛)}
12	9, 10, 11	3eqtr4i 2796	1 ⊢ PetParts = PetErs

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  Vcvv 3454   class class class wbr 5106  {copab 5170   E cep 5554  ^◡ccnv 5654   ↾ cres 5657   ⋉ cxrn 39297   ≀ ccoss 39306   Rels crels 39308   Ers cers 39331   PetErs cpeters 39333   CoMembErs ccomembers 39335   Parts cparts 39346   MembParts cmembparts 39348   PetParts cpetparts 39350

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 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772

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-eu 2597  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 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-eprel 5555  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-fo 6561  df-fv 6563  df-1st 8022  df-2nd 8023  df-ec 8744  df-qs 8748  df-xrn 39503  df-rels 39563  df-coss 39624  df-coels 39625  df-ssr 39701  df-refs 39713  df-refrels 39714  df-refrel 39715  df-cnvrefs 39728  df-cnvrefrels 39729  df-cnvrefrel 39730  df-syms 39745  df-symrels 39746  df-symrel 39747  df-trs 39779  df-trrels 39780  df-trrel 39781  df-eqvrels 39791  df-eqvrel 39792  df-coeleqvrel 39794  df-dmqss 39845  df-dmqs 39846  df-ers 39871  df-erALTV 39872  df-comembers 39873  df-comember 39874  df-funALTV 39890  df-disjss 39911  df-disjs 39912  df-disjALTV 39913  df-eldisj 39915  df-parts 39991  df-part 39992  df-membparts 39993  df-membpart 39994  df-petparts 40091  df-peters 40092

This theorem is used by:  pets2eq  40100