Theorem dfpetparts2 39175

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 39145 / dfdisjs7 39146): 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 5779	. . . 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 38684	. . . . 5 ⊢ BlockLiftFix = {⟨𝑟, 𝑛⟩ ∣ (dom (𝑟 ⋉ (◡ E ↾ 𝑛)) / (𝑟 ⋉ (◡ E ↾ 𝑛))) = 𝑛}
3	2	ineq2i 4170	. . . 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 38634	. . . . . . 7 ⊢ ((𝑛 ∈ V ∧ 𝑟 ∈ V) → (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ V)
5	4	el2v 3448	. . . . . 6 ⊢ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ V
6		brparts2 39078	. . . . . . 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 38432	. . . . . 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 5166	. . . 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 4170	. 2 ⊢ (( Rels × MembParts ) ∩ {⟨𝑟, 𝑛⟩ ∣ (𝑟 ⋉ (◡ E ↾ 𝑛)) Parts 𝑛}) = (( Rels × MembParts ) ∩ ({⟨𝑟, 𝑛⟩ ∣ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ Disjs } ∩ BlockLiftFix ))
12		inopab 5779	. . 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 5631	. . . 4 ⊢ ( Rels × MembParts ) = {⟨𝑟, 𝑛⟩ ∣ (𝑟 ∈ Rels ∧ 𝑛 ∈ MembParts )}
14	13	ineq1i 4169	. . 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 39171	. . 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 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 2770	1 ⊢ PetParts = ((( Rels × MembParts ) ∩ {⟨𝑟, 𝑛⟩ ∣ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ Disjs }) ∩ BlockLiftFix )

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3441   ∩ cin 3901   class class class wbr 5099  {copab 5161   E cep 5524   × cxp 5623  ^◡ccnv 5624  dom cdm 5625   ↾ cres 5627   / cqs 8636   ⋉ cxrn 38377   BlockLiftFix cblockliftfix 38384   Rels crels 38388   Disjs cdisjs 38421   Parts cparts 38426   MembParts cmembparts 38428   PetParts cpetparts 38430

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 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

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 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-eprel 5525  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-fo 6499  df-fv 6501  df-1st 7935  df-2nd 7936  df-ec 8639  df-qs 8643  df-xrn 38583  df-blockliftfix 38684  df-dmqss 38925  df-parts 39071  df-petparts 39171

This theorem is referenced by:  dfpet2parts2  39176