Theorem dfpetparts2 39476

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 39446 / dfdisjs7 39447): 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 5804	. . . 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 38985	. . . . 5 ⊢ BlockLiftFix = {⟨𝑟, 𝑛⟩ ∣ (dom (𝑟 ⋉ (◡ E ↾ 𝑛)) / (𝑟 ⋉ (◡ E ↾ 𝑛))) = 𝑛}
3	2	ineq2i 4171	. . . 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 38935	. . . . . . 7 ⊢ ((𝑛 ∈ V ∧ 𝑟 ∈ V) → (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ V)
5	4	el2v 3463	. . . . . 6 ⊢ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ V
6		brparts2 39379	. . . . . . 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 38733	. . . . . 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 5169	. . . 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 2798	. . 3 ⊢ {⟨𝑟, 𝑛⟩ ∣ (𝑟 ⋉ (◡ E ↾ 𝑛)) Parts 𝑛} = ({⟨𝑟, 𝑛⟩ ∣ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ Disjs } ∩ BlockLiftFix )
11	10	ineq2i 4171	. 2 ⊢ (( Rels × MembParts ) ∩ {⟨𝑟, 𝑛⟩ ∣ (𝑟 ⋉ (◡ E ↾ 𝑛)) Parts 𝑛}) = (( Rels × MembParts ) ∩ ({⟨𝑟, 𝑛⟩ ∣ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ Disjs } ∩ BlockLiftFix ))
12		inopab 5804	. . 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 5655	. . . 4 ⊢ ( Rels × MembParts ) = {⟨𝑟, 𝑛⟩ ∣ (𝑟 ∈ Rels ∧ 𝑛 ∈ MembParts )}
14	13	ineq1i 4170	. . 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 39472	. . 3 ⊢ PetParts = {⟨𝑟, 𝑛⟩ ∣ ((𝑟 ∈ Rels ∧ 𝑛 ∈ MembParts ) ∧ (𝑟 ⋉ (◡ E ↾ 𝑛)) Parts 𝑛)}
16	12, 14, 15	3eqtr4ri 2798	. 2 ⊢ PetParts = (( Rels × MembParts ) ∩ {⟨𝑟, 𝑛⟩ ∣ (𝑟 ⋉ (◡ E ↾ 𝑛)) Parts 𝑛})
17		inass 4181	. 2 ⊢ ((( Rels × MembParts ) ∩ {⟨𝑟, 𝑛⟩ ∣ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ Disjs }) ∩ BlockLiftFix ) = (( Rels × MembParts ) ∩ ({⟨𝑟, 𝑛⟩ ∣ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ Disjs } ∩ BlockLiftFix ))
18	11, 16, 17	3eqtr4i 2797	1 ⊢ PetParts = ((( Rels × MembParts ) ∩ {⟨𝑟, 𝑛⟩ ∣ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ Disjs }) ∩ BlockLiftFix )

Syntax hints:   ↔ wb 208   ∧ wa 399   = wceq 1562   ∈ wcel 2144  Vcvv 3456   ∩ cin 3905   class class class wbr 5102  {copab 5164   E cep 5548   × cxp 5647  ^◡ccnv 5648  dom cdm 5649   ↾ cres 5651   / cqs 8679   ⋉ cxrn 38678   BlockLiftFix cblockliftfix 38685   Rels crels 38689   Disjs cdisjs 38722   Parts cparts 38727   MembParts cmembparts 38729   PetParts cpetparts 38731

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-eprel 5549  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-fo 6529  df-fv 6531  df-1st 7972  df-2nd 7973  df-ec 8682  df-qs 8686  df-xrn 38884  df-blockliftfix 38985  df-dmqss 39226  df-parts 39372  df-petparts 39472

This theorem is referenced by:  dfpet2parts2  39477