Theorem petseq 39358

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 39338 plays for carriers: mpets 39338 identifies MembParts with CoMembErs at the membership-carrier level, while petseq 39358 identifies the corresponding span-level predicates built from Parts and Ers.

Unlike the earlier broad pets 39348, the bridge used here is the type-safe span theorem typesafepets 39357, 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 39357	. . . . . . 7 ⊢ ((𝑛 ∈ MembParts ∧ 𝑟 ∈ V) → ((𝑟 ⋉ (◡ E ↾ 𝑛)) Parts 𝑛 ↔ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛))
2	1	elvd 3439	. . . . . 6 ⊢ (𝑛 ∈ MembParts → ((𝑟 ⋉ (◡ E ↾ 𝑛)) Parts 𝑛 ↔ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛))
3	2	adantl 483	. . . . 5 ⊢ ((𝑟 ∈ Rels ∧ 𝑛 ∈ MembParts ) → ((𝑟 ⋉ (◡ E ↾ 𝑛)) Parts 𝑛 ↔ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛))
4	3	pm5.32i 580	. . . 4 ⊢ (((𝑟 ∈ Rels ∧ 𝑛 ∈ MembParts ) ∧ (𝑟 ⋉ (◡ E ↾ 𝑛)) Parts 𝑛) ↔ ((𝑟 ∈ Rels ∧ 𝑛 ∈ MembParts ) ∧ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛))
5		mpets 39338	. . . . . 6 ⊢ MembParts = CoMembErs
6	5	eleq2i 2833	. . . . 5 ⊢ (𝑛 ∈ MembParts ↔ 𝑛 ∈ CoMembErs )
7	6	anbi2i 630	. . . 4 ⊢ ((𝑟 ∈ Rels ∧ 𝑛 ∈ MembParts ) ↔ (𝑟 ∈ Rels ∧ 𝑛 ∈ CoMembErs ))
8	4, 7	bianbi 634	. . 3 ⊢ (((𝑟 ∈ Rels ∧ 𝑛 ∈ MembParts ) ∧ (𝑟 ⋉ (◡ E ↾ 𝑛)) Parts 𝑛) ↔ ((𝑟 ∈ Rels ∧ 𝑛 ∈ CoMembErs ) ∧ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛))
9	8	opabbii 5142	. 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 39350	. 2 ⊢ PetParts = {⟨𝑟, 𝑛⟩ ∣ ((𝑟 ∈ Rels ∧ 𝑛 ∈ MembParts ) ∧ (𝑟 ⋉ (◡ E ↾ 𝑛)) Parts 𝑛)}
11		df-peters 39351	. 2 ⊢ PetErs = {⟨𝑟, 𝑛⟩ ∣ ((𝑟 ∈ Rels ∧ 𝑛 ∈ CoMembErs ) ∧ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛)}
12	9, 10, 11	3eqtr4i 2774	1 ⊢ PetParts = PetErs

Syntax hints:   ↔ wb 208   ∧ wa 397   = wceq 1548   ∈ wcel 2121  Vcvv 3433   class class class wbr 5075  {copab 5137   E cep 5520  ^◡ccnv 5620   ↾ cres 5623   ⋉ cxrn 38556   ≀ ccoss 38565   Rels crels 38567   Ers cers 38590   PetErs cpeters 38592   CoMembErs ccomembers 38594   Parts cparts 38605   MembParts cmembparts 38607   PetParts cpetparts 38609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rmo 3346  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-eprel 5521  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-fo 6495  df-fv 6497  df-1st 7935  df-2nd 7936  df-ec 8639  df-qs 8643  df-xrn 38762  df-rels 38822  df-coss 38883  df-coels 38884  df-ssr 38960  df-refs 38972  df-refrels 38973  df-refrel 38974  df-cnvrefs 38987  df-cnvrefrels 38988  df-cnvrefrel 38989  df-syms 39004  df-symrels 39005  df-symrel 39006  df-trs 39038  df-trrels 39039  df-trrel 39040  df-eqvrels 39050  df-eqvrel 39051  df-coeleqvrel 39053  df-dmqss 39104  df-dmqs 39105  df-ers 39130  df-erALTV 39131  df-comembers 39132  df-comember 39133  df-funALTV 39149  df-disjss 39170  df-disjs 39171  df-disjALTV 39172  df-eldisj 39174  df-parts 39250  df-part 39251  df-membparts 39252  df-membpart 39253  df-petparts 39350  df-peters 39351

This theorem is referenced by:  pets2eq  39359