Theorem typesafepets 39652

Description: Type-safe pets 39643 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 39643: 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 39653), in complete parallel with the membership bridge mpets 39633. This theorem is intentionally not the definition of PetParts; it is the bridge used by petseq 39653 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 39643	1 ⊢ ((𝐴 ∈ MembParts ∧ 𝑅 ∈ 𝑉) → ((𝑅 ⋉ (◡ E ↾ 𝐴)) Parts 𝐴 ↔ ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) Ers 𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   ∈ wcel 2142   class class class wbr 5108   E cep 5559  ^◡ccnv 5659   ↾ cres 5662   ⋉ cxrn 38851   ≀ ccoss 38860   Ers cers 38885   Parts cparts 38900   MembParts cmembparts 38902

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-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-dmqss 39399  df-dmqs 39400  df-ers 39425  df-erALTV 39426  df-funALTV 39444  df-disjss 39465  df-disjs 39466  df-disjALTV 39467  df-eldisj 39469  df-parts 39545  df-part 39546

This theorem is used by:  petseq  39653