Theorem dfpetparts2 39252

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 39222 / dfdisjs7 39223): 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 5788	. . . 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 38761	. . . . 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 38711	. . . . . . 7 ⊢ ((𝑛 ∈ V ∧ 𝑟 ∈ V) → (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ V)
5	4	el2v 3449	. . . . . 6 ⊢ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ V
6		brparts2 39155	. . . . . . 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 38509	. . . . . 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 5167	. . . 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 2771	. . 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 5788	. . 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 5640	. . . 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 39248	. . 3 ⊢ PetParts = {⟨𝑟, 𝑛⟩ ∣ ((𝑟 ∈ Rels ∧ 𝑛 ∈ MembParts ) ∧ (𝑟 ⋉ (◡ E ↾ 𝑛)) Parts 𝑛)}
16	12, 14, 15	3eqtr4ri 2771	. 2 ⊢ PetParts = (( Rels × MembParts ) ∩ {⟨𝑟, 𝑛⟩ ∣ (𝑟 ⋉ (◡ E ↾ 𝑛)) Parts 𝑛})
17		inass 4182	. 2 ⊢ ((( Rels × MembParts ) ∩ {⟨𝑟, 𝑛⟩ ∣ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ Disjs }) ∩ BlockLiftFix ) = (( Rels × MembParts ) ∩ ({⟨𝑟, 𝑛⟩ ∣ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ Disjs } ∩ BlockLiftFix ))
18	11, 16, 17	3eqtr4i 2770	1 ⊢ PetParts = ((( Rels × MembParts ) ∩ {⟨𝑟, 𝑛⟩ ∣ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ Disjs }) ∩ BlockLiftFix )

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3442   ∩ cin 3902   class class class wbr 5100  {copab 5162   E cep 5533   × cxp 5632  ^◡ccnv 5633  dom cdm 5634   ↾ cres 5636   / cqs 8646   ⋉ cxrn 38454   BlockLiftFix cblockliftfix 38461   Rels crels 38465   Disjs cdisjs 38498   Parts cparts 38503   MembParts cmembparts 38505   PetParts cpetparts 38507

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-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-blockliftfix 38761  df-dmqss 39002  df-parts 39148  df-petparts 39248

This theorem is referenced by:  dfpet2parts2  39253