Theorem typesafepets 39827

Description: Type-safe pets 39818 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 39818: 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 39828), in complete parallel with the membership bridge mpets 39808. This theorem is intentionally not the definition of PetParts; it is the bridge used by petseq 39828 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 39818	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 401   ∈ wcel 2145   class class class wbr 5102   E cep 5546  ^◡ccnv 5646   ↾ cres 5649   ⋉ cxrn 39026   ≀ ccoss 39035   Ers cers 39060   Parts cparts 39075   MembParts cmembparts 39077

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  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-dif 3901  df-un 3903  df-in 3905  df-ss 3915  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 5542  df-eprel 5547  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-fo 6533  df-fv 6535  df-1st 7984  df-2nd 7985  df-ec 8697  df-qs 8701  df-xrn 39232  df-rels 39292  df-coss 39353  df-ssr 39430  df-refs 39442  df-refrels 39443  df-refrel 39444  df-cnvrefs 39457  df-cnvrefrels 39458  df-cnvrefrel 39459  df-syms 39474  df-symrels 39475  df-symrel 39476  df-trs 39508  df-trrels 39509  df-trrel 39510  df-eqvrels 39520  df-eqvrel 39521  df-dmqss 39574  df-dmqs 39575  df-ers 39600  df-erALTV 39601  df-funALTV 39619  df-disjss 39640  df-disjs 39641  df-disjALTV 39642  df-eldisj 39644  df-parts 39720  df-part 39721

This theorem is used by:  petseq  39828