Theorem petseq 39653

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 39633 plays for carriers: mpets 39633 identifies MembParts with CoMembErs at the membership-carrier level, while petseq 39653 identifies the corresponding span-level predicates built from Parts and Ers.

Unlike the earlier broad pets 39643, the bridge used here is the type-safe span theorem typesafepets 39652, 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 39652	. . . . . . 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 486	. . . . 5 ⊢ ((𝑟 ∈ Rels ∧ 𝑛 ∈ MembParts ) → ((𝑟 ⋉ (◡ E ↾ 𝑛)) Parts 𝑛 ↔ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛))
4	3	pm5.32i 584	. . . 4 ⊢ (((𝑟 ∈ Rels ∧ 𝑛 ∈ MembParts ) ∧ (𝑟 ⋉ (◡ E ↾ 𝑛)) Parts 𝑛) ↔ ((𝑟 ∈ Rels ∧ 𝑛 ∈ MembParts ) ∧ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛))
5		mpets 39633	. . . . . 6 ⊢ MembParts = CoMembErs
6	5	eleq2i 2854	. . . . 5 ⊢ (𝑛 ∈ MembParts ↔ 𝑛 ∈ CoMembErs )
7	6	anbi2i 634	. . . 4 ⊢ ((𝑟 ∈ Rels ∧ 𝑛 ∈ MembParts ) ↔ (𝑟 ∈ Rels ∧ 𝑛 ∈ CoMembErs ))
8	4, 7	bianbi 638	. . 3 ⊢ (((𝑟 ∈ Rels ∧ 𝑛 ∈ MembParts ) ∧ (𝑟 ⋉ (◡ E ↾ 𝑛)) Parts 𝑛) ↔ ((𝑟 ∈ Rels ∧ 𝑛 ∈ CoMembErs ) ∧ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛))
9	8	opabbii 5177	. 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 39645	. 2 ⊢ PetParts = {⟨𝑟, 𝑛⟩ ∣ ((𝑟 ∈ Rels ∧ 𝑛 ∈ MembParts ) ∧ (𝑟 ⋉ (◡ E ↾ 𝑛)) Parts 𝑛)}
11		df-peters 39646	. 2 ⊢ PetErs = {⟨𝑟, 𝑛⟩ ∣ ((𝑟 ∈ Rels ∧ 𝑛 ∈ CoMembErs ) ∧ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛)}
12	9, 10, 11	3eqtr4i 2795	1 ⊢ PetParts = PetErs

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 400   = wceq 1569   ∈ wcel 2142  Vcvv 3454   class class class wbr 5108  {copab 5172   E cep 5559  ^◡ccnv 5659   ↾ cres 5662   ⋉ cxrn 38851   ≀ ccoss 38860   Rels crels 38862   Ers cers 38885   PetErs cpeters 38887   CoMembErs ccomembers 38889   Parts cparts 38900   MembParts cmembparts 38902   PetParts cpetparts 38904

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3368  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-eprel 5560  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fo 6542  df-fv 6544  df-1st 7984  df-2nd 7985  df-ec 8694  df-qs 8698  df-xrn 39057  df-rels 39117  df-coss 39178  df-coels 39179  df-ssr 39255  df-refs 39267  df-refrels 39268  df-refrel 39269  df-cnvrefs 39282  df-cnvrefrels 39283  df-cnvrefrel 39284  df-syms 39299  df-symrels 39300  df-symrel 39301  df-trs 39333  df-trrels 39334  df-trrel 39335  df-eqvrels 39345  df-eqvrel 39346  df-coeleqvrel 39348  df-dmqss 39399  df-dmqs 39400  df-ers 39425  df-erALTV 39426  df-comembers 39427  df-comember 39428  df-funALTV 39444  df-disjss 39465  df-disjs 39466  df-disjALTV 39467  df-eldisj 39469  df-parts 39545  df-part 39546  df-membparts 39547  df-membpart 39548  df-petparts 39645  df-peters 39646

This theorem is used by:  pets2eq  39654