Theorem dfpeters2 40098

Description: Alternate definition of PetErs in fully modular form.

This expands the Ers 𝑛 predicate into:

(i) a typedness module ( Rels × CoMembErs ),

(ii) an equivalence module for the coset relation ≀ (𝑟 ⋉ (^◡ E ↾ 𝑛)) ∈ EqvRels,

(iii) the corresponding quotient-carrier (domain quotient) equation dom ≀ (...) / ≀ (...) = 𝑛.

This is the equivalence-side counterpart of the modular decomposition dfpetparts2 40096 on the partition side. (Contributed by Peter Mazsa, 25-Feb-2026.)

Assertion

Ref	Expression
dfpeters2	⊢ PetErs = ((( Rels × CoMembErs ) ∩ {⟨𝑟, 𝑛⟩ ∣ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ EqvRels }) ∩ {⟨𝑟, 𝑛⟩ ∣ (dom ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) / ≀ (𝑟 ⋉ (◡ E ↾ 𝑛))) = 𝑛})

Distinct variable group:   𝑛,𝑟

Proof of Theorem dfpeters2

Step	Hyp	Ref	Expression
1		brers 39876	. . . . . . 7 ⊢ (𝑛 ∈ V → ( ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛 ↔ ( ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ EqvRels ∧ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) DomainQss 𝑛)))
2	1	elv 3459	. . . . . 6 ⊢ ( ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛 ↔ ( ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ EqvRels ∧ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) DomainQss 𝑛))
3		1cossxrncnvepresex 39636	. . . . . . . . . 10 ⊢ ((𝑛 ∈ V ∧ 𝑟 ∈ V) → ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ V)
4	3	el2v 3461	. . . . . . . . 9 ⊢ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ V
5		brdmqss 39854	. . . . . . . . 9 ⊢ ((𝑛 ∈ V ∧ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ V) → ( ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) DomainQss 𝑛 ↔ (dom ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) / ≀ (𝑟 ⋉ (◡ E ↾ 𝑛))) = 𝑛))
6	4, 5	mpan2 704	. . . . . . . 8 ⊢ (𝑛 ∈ V → ( ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) DomainQss 𝑛 ↔ (dom ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) / ≀ (𝑟 ⋉ (◡ E ↾ 𝑛))) = 𝑛))
7	6	elv 3459	. . . . . . 7 ⊢ ( ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) DomainQss 𝑛 ↔ (dom ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) / ≀ (𝑟 ⋉ (◡ E ↾ 𝑛))) = 𝑛)
8	7	anbi2i 635	. . . . . 6 ⊢ (( ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ EqvRels ∧ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) DomainQss 𝑛) ↔ ( ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ EqvRels ∧ (dom ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) / ≀ (𝑟 ⋉ (◡ E ↾ 𝑛))) = 𝑛))
9	2, 8	bitri 278	. . . . 5 ⊢ ( ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛 ↔ ( ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ EqvRels ∧ (dom ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) / ≀ (𝑟 ⋉ (◡ E ↾ 𝑛))) = 𝑛))
10	9	opabbii 5175	. . . 4 ⊢ {⟨𝑟, 𝑛⟩ ∣ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛} = {⟨𝑟, 𝑛⟩ ∣ ( ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ EqvRels ∧ (dom ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) / ≀ (𝑟 ⋉ (◡ E ↾ 𝑛))) = 𝑛)}
11		inopab 5813	. . . 4 ⊢ ({⟨𝑟, 𝑛⟩ ∣ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ EqvRels } ∩ {⟨𝑟, 𝑛⟩ ∣ (dom ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) / ≀ (𝑟 ⋉ (◡ E ↾ 𝑛))) = 𝑛}) = {⟨𝑟, 𝑛⟩ ∣ ( ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ EqvRels ∧ (dom ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) / ≀ (𝑟 ⋉ (◡ E ↾ 𝑛))) = 𝑛)}
12	10, 11	eqtr4i 2789	. . 3 ⊢ {⟨𝑟, 𝑛⟩ ∣ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛} = ({⟨𝑟, 𝑛⟩ ∣ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ EqvRels } ∩ {⟨𝑟, 𝑛⟩ ∣ (dom ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) / ≀ (𝑟 ⋉ (◡ E ↾ 𝑛))) = 𝑛})
13	12	ineq2i 4117	. 2 ⊢ (( Rels × CoMembErs ) ∩ {⟨𝑟, 𝑛⟩ ∣ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛}) = (( Rels × CoMembErs ) ∩ ({⟨𝑟, 𝑛⟩ ∣ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ EqvRels } ∩ {⟨𝑟, 𝑛⟩ ∣ (dom ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) / ≀ (𝑟 ⋉ (◡ E ↾ 𝑛))) = 𝑛}))
14		inopab 5813	. . 3 ⊢ ({⟨𝑟, 𝑛⟩ ∣ (𝑟 ∈ Rels ∧ 𝑛 ∈ CoMembErs )} ∩ {⟨𝑟, 𝑛⟩ ∣ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛}) = {⟨𝑟, 𝑛⟩ ∣ ((𝑟 ∈ Rels ∧ 𝑛 ∈ CoMembErs ) ∧ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛)}
15		df-xp 5661	. . . 4 ⊢ ( Rels × CoMembErs ) = {⟨𝑟, 𝑛⟩ ∣ (𝑟 ∈ Rels ∧ 𝑛 ∈ CoMembErs )}
16	15	ineq1i 4116	. . 3 ⊢ (( Rels × CoMembErs ) ∩ {⟨𝑟, 𝑛⟩ ∣ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛}) = ({⟨𝑟, 𝑛⟩ ∣ (𝑟 ∈ Rels ∧ 𝑛 ∈ CoMembErs )} ∩ {⟨𝑟, 𝑛⟩ ∣ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛})
17		df-peters 40093	. . 3 ⊢ PetErs = {⟨𝑟, 𝑛⟩ ∣ ((𝑟 ∈ Rels ∧ 𝑛 ∈ CoMembErs ) ∧ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛)}
18	14, 16, 17	3eqtr4ri 2797	. 2 ⊢ PetErs = (( Rels × CoMembErs ) ∩ {⟨𝑟, 𝑛⟩ ∣ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛})
19		inass 4127	. 2 ⊢ ((( Rels × CoMembErs ) ∩ {⟨𝑟, 𝑛⟩ ∣ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ EqvRels }) ∩ {⟨𝑟, 𝑛⟩ ∣ (dom ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) / ≀ (𝑟 ⋉ (◡ E ↾ 𝑛))) = 𝑛}) = (( Rels × CoMembErs ) ∩ ({⟨𝑟, 𝑛⟩ ∣ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ EqvRels } ∩ {⟨𝑟, 𝑛⟩ ∣ (dom ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) / ≀ (𝑟 ⋉ (◡ E ↾ 𝑛))) = 𝑛}))
20	13, 18, 19	3eqtr4i 2796	1 ⊢ PetErs = ((( Rels × CoMembErs ) ∩ {⟨𝑟, 𝑛⟩ ∣ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ EqvRels }) ∩ {⟨𝑟, 𝑛⟩ ∣ (dom ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) / ≀ (𝑟 ⋉ (◡ E ↾ 𝑛))) = 𝑛})

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  Vcvv 3454   ∩ cin 4092   class class class wbr 5106  {copab 5170   E cep 5554   × cxp 5653  ^◡ccnv 5654  dom cdm 5655   ↾ cres 5657   / cqs 8744   ⋉ cxrn 39298   ≀ ccoss 39307   Rels crels 39309   EqvRels ceqvrels 39323   DomainQss cdmqss 39330   Ers cers 39332   PetErs cpeters 39334   CoMembErs ccomembers 39336

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-eprel 5555  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-fo 6564  df-fv 6566  df-1st 8025  df-2nd 8026  df-ec 8747  df-qs 8751  df-xrn 39504  df-coss 39625  df-dmqss 39846  df-ers 39872  df-peters 40093

This theorem is used by: (None)