Theorem dfpetparts2 39571

Description: Alternate definition of PetParts as typedness + disjoint-span + block-lift equilibrium.

This theorem is the key modularization step. It decomposes PetParts into the intersection of three orthogonal modules:

(T) typedness: ⟨𝑟, 𝑛⟩ ∈ ( Rels × MembParts ),

(D) disjoint-span: (𝑟 ⋉ (^◡ E ↾ 𝑛)) ∈ Disjs,

(E) semantic equilibrium: ⟨𝑟, 𝑛⟩ ∈ BlockLiftFix, i.e. the carrier 𝑛 is a fixpoint of the induced block-generation operator.

Conceptually, (D) provides the disjointness/quotient discipline for the lifted span, while (E) prevents hidden carrier drift (refinement or coarsening of what counts as a block) by enforcing the fixpoint equation. The point of this theorem is that these constraints can be imposed and reused independently by later constructions, while their intersection recovers the intended Parts-based notion.

This mirrors the internal packaging of Disjs (see dfdisjs6 39541 / dfdisjs7 39542): for disjoint relations, the "map layer + carrier layer" decomposition is internal via QMap and ElDisjs; for PetParts, the carrier 𝑛 is an external parameter, so the additional carrier stability must be factored explicitly as BlockLiftFix. (Contributed by Peter Mazsa, 20-Feb-2026.) (Revised by Peter Mazsa, 25-Feb-2026.)

Assertion

Ref	Expression
dfpetparts2	⊢ PetParts = ((( Rels × MembParts ) ∩ {⟨𝑟, 𝑛⟩ ∣ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ Disjs }) ∩ BlockLiftFix )

Distinct variable group:   𝑛,𝑟

Proof of Theorem dfpetparts2

Step	Hyp	Ref	Expression
1		inopab 5820	. . . 4 ⊢ ({⟨𝑟, 𝑛⟩ ∣ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ Disjs } ∩ {⟨𝑟, 𝑛⟩ ∣ (dom (𝑟 ⋉ (◡ E ↾ 𝑛)) / (𝑟 ⋉ (◡ E ↾ 𝑛))) = 𝑛}) = {⟨𝑟, 𝑛⟩ ∣ ((𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ Disjs ∧ (dom (𝑟 ⋉ (◡ E ↾ 𝑛)) / (𝑟 ⋉ (◡ E ↾ 𝑛))) = 𝑛)}
2		df-blockliftfix 39080	. . . . 5 ⊢ BlockLiftFix = {⟨𝑟, 𝑛⟩ ∣ (dom (𝑟 ⋉ (◡ E ↾ 𝑛)) / (𝑟 ⋉ (◡ E ↾ 𝑛))) = 𝑛}
3	2	ineq2i 4178	. . . 4 ⊢ ({⟨𝑟, 𝑛⟩ ∣ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ Disjs } ∩ BlockLiftFix ) = ({⟨𝑟, 𝑛⟩ ∣ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ Disjs } ∩ {⟨𝑟, 𝑛⟩ ∣ (dom (𝑟 ⋉ (◡ E ↾ 𝑛)) / (𝑟 ⋉ (◡ E ↾ 𝑛))) = 𝑛})
4		xrncnvepresex 39030	. . . . . . 7 ⊢ ((𝑛 ∈ V ∧ 𝑟 ∈ V) → (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ V)
5	4	el2v 3469	. . . . . 6 ⊢ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ V
6		brparts2 39474	. . . . . . 7 ⊢ ((𝑛 ∈ V ∧ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ V) → ((𝑟 ⋉ (◡ E ↾ 𝑛)) Parts 𝑛 ↔ ((𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ Disjs ∧ (dom (𝑟 ⋉ (◡ E ↾ 𝑛)) / (𝑟 ⋉ (◡ E ↾ 𝑛))) = 𝑛)))
7	6	el2v1 38828	. . . . . 6 ⊢ ((𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ V → ((𝑟 ⋉ (◡ E ↾ 𝑛)) Parts 𝑛 ↔ ((𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ Disjs ∧ (dom (𝑟 ⋉ (◡ E ↾ 𝑛)) / (𝑟 ⋉ (◡ E ↾ 𝑛))) = 𝑛)))
8	5, 7	ax-mp 5	. . . . 5 ⊢ ((𝑟 ⋉ (◡ E ↾ 𝑛)) Parts 𝑛 ↔ ((𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ Disjs ∧ (dom (𝑟 ⋉ (◡ E ↾ 𝑛)) / (𝑟 ⋉ (◡ E ↾ 𝑛))) = 𝑛))
9	8	opabbii 5183	. . . 4 ⊢ {⟨𝑟, 𝑛⟩ ∣ (𝑟 ⋉ (◡ E ↾ 𝑛)) Parts 𝑛} = {⟨𝑟, 𝑛⟩ ∣ ((𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ Disjs ∧ (dom (𝑟 ⋉ (◡ E ↾ 𝑛)) / (𝑟 ⋉ (◡ E ↾ 𝑛))) = 𝑛)}
10	1, 3, 9	3eqtr4ri 2804	. . 3 ⊢ {⟨𝑟, 𝑛⟩ ∣ (𝑟 ⋉ (◡ E ↾ 𝑛)) Parts 𝑛} = ({⟨𝑟, 𝑛⟩ ∣ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ Disjs } ∩ BlockLiftFix )
11	10	ineq2i 4178	. 2 ⊢ (( Rels × MembParts ) ∩ {⟨𝑟, 𝑛⟩ ∣ (𝑟 ⋉ (◡ E ↾ 𝑛)) Parts 𝑛}) = (( Rels × MembParts ) ∩ ({⟨𝑟, 𝑛⟩ ∣ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ Disjs } ∩ BlockLiftFix ))
12		inopab 5820	. . 3 ⊢ ({⟨𝑟, 𝑛⟩ ∣ (𝑟 ∈ Rels ∧ 𝑛 ∈ MembParts )} ∩ {⟨𝑟, 𝑛⟩ ∣ (𝑟 ⋉ (◡ E ↾ 𝑛)) Parts 𝑛}) = {⟨𝑟, 𝑛⟩ ∣ ((𝑟 ∈ Rels ∧ 𝑛 ∈ MembParts ) ∧ (𝑟 ⋉ (◡ E ↾ 𝑛)) Parts 𝑛)}
13		df-xp 5671	. . . 4 ⊢ ( Rels × MembParts ) = {⟨𝑟, 𝑛⟩ ∣ (𝑟 ∈ Rels ∧ 𝑛 ∈ MembParts )}
14	13	ineq1i 4177	. . 3 ⊢ (( Rels × MembParts ) ∩ {⟨𝑟, 𝑛⟩ ∣ (𝑟 ⋉ (◡ E ↾ 𝑛)) Parts 𝑛}) = ({⟨𝑟, 𝑛⟩ ∣ (𝑟 ∈ Rels ∧ 𝑛 ∈ MembParts )} ∩ {⟨𝑟, 𝑛⟩ ∣ (𝑟 ⋉ (◡ E ↾ 𝑛)) Parts 𝑛})
15		df-petparts 39567	. . 3 ⊢ PetParts = {⟨𝑟, 𝑛⟩ ∣ ((𝑟 ∈ Rels ∧ 𝑛 ∈ MembParts ) ∧ (𝑟 ⋉ (◡ E ↾ 𝑛)) Parts 𝑛)}
16	12, 14, 15	3eqtr4ri 2804	. 2 ⊢ PetParts = (( Rels × MembParts ) ∩ {⟨𝑟, 𝑛⟩ ∣ (𝑟 ⋉ (◡ E ↾ 𝑛)) Parts 𝑛})
17		inass 4188	. 2 ⊢ ((( Rels × MembParts ) ∩ {⟨𝑟, 𝑛⟩ ∣ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ Disjs }) ∩ BlockLiftFix ) = (( Rels × MembParts ) ∩ ({⟨𝑟, 𝑛⟩ ∣ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ Disjs } ∩ BlockLiftFix ))
18	11, 16, 17	3eqtr4i 2803	1 ⊢ PetParts = ((( Rels × MembParts ) ∩ {⟨𝑟, 𝑛⟩ ∣ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ Disjs }) ∩ BlockLiftFix )

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1568   ∈ wcel 2150  Vcvv 3462   ∩ cin 3912   class class class wbr 5114  {copab 5178   E cep 5564   × cxp 5663  ^◡ccnv 5664  dom cdm 5665   ↾ cres 5667   / cqs 8696   ⋉ cxrn 38773   BlockLiftFix cblockliftfix 38780   Rels crels 38784   Disjs cdisjs 38817   Parts cparts 38822   MembParts cmembparts 38824   PetParts cpetparts 38826

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-10 2183  ax-11 2199  ax-12 2220  ax-ext 2742  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5340  ax-pr 5408  ax-un 7736

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2099  df-mo 2574  df-eu 2604  df-clab 2749  df-cleq 2762  df-clel 2845  df-nfc 2919  df-ne 2966  df-ral 3087  df-rex 3097  df-rab 3424  df-v 3464  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5560  df-eprel 5565  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-fo 6546  df-fv 6548  df-1st 7989  df-2nd 7990  df-ec 8699  df-qs 8703  df-xrn 38979  df-blockliftfix 39080  df-dmqss 39321  df-parts 39467  df-petparts 39567

This theorem is referenced by:  dfpet2parts2  39572