Theorem typesafepets 39255

Description: Type-safe pets 39246 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 39246: 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 39256), in complete parallel with the membership bridge mpets 39236. This theorem is intentionally not the definition of PetParts; it is the bridge used by petseq 39256 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 39246	1 ⊢ ((𝐴 ∈ MembParts ∧ 𝑅 ∈ 𝑉) → ((𝑅 ⋉ (◡ E ↾ 𝐴)) Parts 𝐴 ↔ ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) Ers 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2114   class class class wbr 5100   E cep 5533  ^◡ccnv 5633   ↾ cres 5636   ⋉ cxrn 38454   ≀ ccoss 38463   Ers cers 38488   Parts cparts 38503   MembParts cmembparts 38505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5529  df-eprel 5534  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-fo 6508  df-fv 6510  df-1st 7945  df-2nd 7946  df-ec 8649  df-qs 8653  df-xrn 38660  df-rels 38720  df-coss 38781  df-ssr 38858  df-refs 38870  df-refrels 38871  df-refrel 38872  df-cnvrefs 38885  df-cnvrefrels 38886  df-cnvrefrel 38887  df-syms 38902  df-symrels 38903  df-symrel 38904  df-trs 38936  df-trrels 38937  df-trrel 38938  df-eqvrels 38948  df-eqvrel 38949  df-dmqss 39002  df-dmqs 39003  df-ers 39028  df-erALTV 39029  df-funALTV 39047  df-disjss 39068  df-disjs 39069  df-disjALTV 39070  df-eldisj 39072  df-parts 39148  df-part 39149

This theorem is referenced by:  petseq  39256