Theorem petseq 39909

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 39889 plays for carriers: mpets 39889 identifies MembParts with CoMembErs at the membership-carrier level, while petseq 39909 identifies the corresponding span-level predicates built from Parts and Ers.

Unlike the earlier broad pets 39899, the bridge used here is the type-safe span theorem typesafepets 39908, 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 39908	. . . . . . 7 ⊢ ((𝑛 ∈ MembParts ∧ 𝑟 ∈ V) → ((𝑟 ⋉ (◡ E ↾ 𝑛)) Parts 𝑛 ↔ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛))
2	1	elvd 3456	. . . . . 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 39889	. . . . . 6 ⊢ MembParts = CoMembErs
6	5	eleq2i 2852	. . . . 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 5171	. 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 39901	. 2 ⊢ PetParts = {⟨𝑟, 𝑛⟩ ∣ ((𝑟 ∈ Rels ∧ 𝑛 ∈ MembParts ) ∧ (𝑟 ⋉ (◡ E ↾ 𝑛)) Parts 𝑛)}
11		df-peters 39902	. 2 ⊢ PetErs = {⟨𝑟, 𝑛⟩ ∣ ((𝑟 ∈ Rels ∧ 𝑛 ∈ CoMembErs ) ∧ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛)}
12	9, 10, 11	3eqtr4i 2793	1 ⊢ PetParts = PetErs

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  Vcvv 3450   class class class wbr 5102  {copab 5166   E cep 5549  ^◡ccnv 5649   ↾ cres 5652   ⋉ cxrn 39107   ≀ ccoss 39116   Rels crels 39118   Ers cers 39141   PetErs cpeters 39143   CoMembErs ccomembers 39145   Parts cparts 39156   MembParts cmembparts 39158   PetParts cpetparts 39160

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 5326  ax-pr 5390  ax-un 7751

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-eu 2594  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 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-eprel 5550  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fo 6544  df-fv 6546  df-1st 8001  df-2nd 8002  df-ec 8719  df-qs 8723  df-xrn 39313  df-rels 39373  df-coss 39434  df-coels 39435  df-ssr 39511  df-refs 39523  df-refrels 39524  df-refrel 39525  df-cnvrefs 39538  df-cnvrefrels 39539  df-cnvrefrel 39540  df-syms 39555  df-symrels 39556  df-symrel 39557  df-trs 39589  df-trrels 39590  df-trrel 39591  df-eqvrels 39601  df-eqvrel 39602  df-coeleqvrel 39604  df-dmqss 39655  df-dmqs 39656  df-ers 39681  df-erALTV 39682  df-comembers 39683  df-comember 39684  df-funALTV 39700  df-disjss 39721  df-disjs 39722  df-disjALTV 39723  df-eldisj 39725  df-parts 39801  df-part 39802  df-membparts 39803  df-membpart 39804  df-petparts 39901  df-peters 39902

This theorem is used by:  pets2eq  39910