Theorem dfpet2parts2 39825

Description: Grade stability applied to the decomposed PetParts modules.

Pet2Parts is obtained by applying the grade-stability operator SucMap ShiftStable (see df-shiftstable 39334) to the modular intersection from dfpetparts2 39824. This makes the two orthogonal stability axes explicit:

(E) semantic stability / equilibrium: BlockLiftFix,

(G) grade stability: SucMap ShiftStable,

assembled on top of typedness and disjoint-span base modules.

This is the principled "extra level" that does not arise for Disjs: disjoint relations already bundle their internal map/carrier consistency via QMap and ElDisjs (see dfdisjs6 39794 / dfdisjs7 39795), while the present construction has an additional external grading axis imposed by the canonical successor map SucMap. (Contributed by Peter Mazsa, 20-Feb-2026.) (Revised by Peter Mazsa, 25-Feb-2026.)

Assertion

Ref	Expression
dfpet2parts2	⊢ Pet2Parts = ( SucMap ShiftStable ((( Rels × MembParts ) ∩ {⟨𝑟, 𝑛⟩ ∣ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ Disjs }) ∩ BlockLiftFix ))

Distinct variable group:   𝑛,𝑟

Proof of Theorem dfpet2parts2

Step	Hyp	Ref	Expression
1		df-pet2parts 39822	. 2 ⊢ Pet2Parts = ( SucMap ShiftStable PetParts )
2		dfpetparts2 39824	. . 3 ⊢ PetParts = ((( Rels × MembParts ) ∩ {⟨𝑟, 𝑛⟩ ∣ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ Disjs }) ∩ BlockLiftFix )
3		shiftstableeq2 39335	. . 3 ⊢ ( PetParts = ((( Rels × MembParts ) ∩ {⟨𝑟, 𝑛⟩ ∣ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ Disjs }) ∩ BlockLiftFix ) → ( SucMap ShiftStable PetParts ) = ( SucMap ShiftStable ((( Rels × MembParts ) ∩ {⟨𝑟, 𝑛⟩ ∣ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ Disjs }) ∩ BlockLiftFix )))
4	2, 3	ax-mp 5	. 2 ⊢ ( SucMap ShiftStable PetParts ) = ( SucMap ShiftStable ((( Rels × MembParts ) ∩ {⟨𝑟, 𝑛⟩ ∣ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ Disjs }) ∩ BlockLiftFix ))
5	1, 4	eqtri 2783	1 ⊢ Pet2Parts = ( SucMap ShiftStable ((( Rels × MembParts ) ∩ {⟨𝑟, 𝑛⟩ ∣ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ Disjs }) ∩ BlockLiftFix ))

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2145   ∩ cin 3897  {copab 5166   E cep 5546   × cxp 5645  ^◡ccnv 5646   ↾ cres 5649   ⋉ cxrn 39026   SucMap csucmap 39030   BlockLiftFix cblockliftfix 39033   ShiftStable cshiftstable 39034   Rels crels 39037   Disjs cdisjs 39070   MembParts cmembparts 39077   PetParts cpetparts 39079   Pet2Parts cpet2parts 39080

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-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-blockliftfix 39333  df-shiftstable 39334  df-dmqss 39574  df-parts 39720  df-petparts 39820  df-pet2parts 39822

This theorem is used by: (None)