Theorem typesafepets 39908

Description: Type-safe pets 39899 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 39899: 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 39909), in complete parallel with the membership bridge mpets 39889. This theorem is intentionally not the definition of PetParts; it is the bridge used by petseq 39909 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 39899	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 2144   class class class wbr 5102   E cep 5549  ^◡ccnv 5649   ↾ cres 5652   ⋉ cxrn 39107   ≀ ccoss 39116   Ers cers 39141   Parts cparts 39156   MembParts cmembparts 39158

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-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-dmqss 39655  df-dmqs 39656  df-ers 39681  df-erALTV 39682  df-funALTV 39700  df-disjss 39721  df-disjs 39722  df-disjALTV 39723  df-eldisj 39725  df-parts 39801  df-part 39802

This theorem is used by:  petseq  39909