Theorem typesafepets 39574

Description: Type-safe pets 39565 scheme. On a membership block-carrier 𝐴 ∈ MembParts, the lifted span (𝑅 ⋉ (^◡ E ↾ 𝐴)) yields a generalized partition of 𝐴 iff its coset relation yields an equivalence relation on the same carrier 𝐴. This is the type-safe replacement for the earlier broad pets 39565: it explicitly restricts to carriers where 𝐴 is already known to be a block-family (by MembParts). That removes the standard type-safety objection ("are you equating a quotient-carrier of blocks with raw witnesses?") by construction. It is the key bridge used to identify the partition-side and equivalence-side pet classes (petseq 39575), in complete parallel with the membership bridge mpets 39555. This theorem is intentionally not the definition of PetParts; it is the bridge used by petseq 39575 after typedness is enforced by the "Pet*" definitions. (Contributed by Peter Mazsa, 19-Feb-2026.)

Assertion

Ref	Expression
typesafepets	⊢ ((𝐴 ∈ MembParts ∧ 𝑅 ∈ 𝑉) → ((𝑅 ⋉ (◡ E ↾ 𝐴)) Parts 𝐴 ↔ ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) Ers 𝐴))

Proof of Theorem typesafepets

Step	Hyp	Ref	Expression
1		pets 39565	1 ⊢ ((𝐴 ∈ MembParts ∧ 𝑅 ∈ 𝑉) → ((𝑅 ⋉ (◡ E ↾ 𝐴)) Parts 𝐴 ↔ ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) Ers 𝐴))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∈ wcel 2150   class class class wbr 5114   E cep 5564  ^◡ccnv 5664   ↾ cres 5667   ⋉ cxrn 38773   ≀ ccoss 38782   Ers cers 38807   Parts cparts 38822   MembParts cmembparts 38824

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-10 2183  ax-11 2199  ax-12 2220  ax-ext 2742  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5340  ax-pr 5408  ax-un 7736

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2099  df-mo 2574  df-eu 2604  df-clab 2749  df-cleq 2762  df-clel 2845  df-nfc 2919  df-ne 2966  df-ral 3087  df-rex 3097  df-rmo 3376  df-rab 3424  df-v 3464  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5560  df-eprel 5565  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-fo 6546  df-fv 6548  df-1st 7989  df-2nd 7990  df-ec 8699  df-qs 8703  df-xrn 38979  df-rels 39039  df-coss 39100  df-ssr 39177  df-refs 39189  df-refrels 39190  df-refrel 39191  df-cnvrefs 39204  df-cnvrefrels 39205  df-cnvrefrel 39206  df-syms 39221  df-symrels 39222  df-symrel 39223  df-trs 39255  df-trrels 39256  df-trrel 39257  df-eqvrels 39267  df-eqvrel 39268  df-dmqss 39321  df-dmqs 39322  df-ers 39347  df-erALTV 39348  df-funALTV 39366  df-disjss 39387  df-disjs 39388  df-disjALTV 39389  df-eldisj 39391  df-parts 39467  df-part 39468

This theorem is referenced by:  petseq  39575