Theorem dfpeters2 39411

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 39409 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 39189	. . . . . . 7 ⊢ (𝑛 ∈ V → ( ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛 ↔ ( ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ EqvRels ∧ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) DomainQss 𝑛)))
2	1	elv 3449	. . . . . 6 ⊢ ( ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛 ↔ ( ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ EqvRels ∧ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) DomainQss 𝑛))
3		1cossxrncnvepresex 38949	. . . . . . . . . 10 ⊢ ((𝑛 ∈ V ∧ 𝑟 ∈ V) → ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ V)
4	3	el2v 3451	. . . . . . . . 9 ⊢ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ V
5		brdmqss 39167	. . . . . . . . 9 ⊢ ((𝑛 ∈ V ∧ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ V) → ( ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) DomainQss 𝑛 ↔ (dom ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) / ≀ (𝑟 ⋉ (◡ E ↾ 𝑛))) = 𝑛))
6	4, 5	mpan2 699	. . . . . . . 8 ⊢ (𝑛 ∈ V → ( ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) DomainQss 𝑛 ↔ (dom ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) / ≀ (𝑟 ⋉ (◡ E ↾ 𝑛))) = 𝑛))
7	6	elv 3449	. . . . . . 7 ⊢ ( ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) DomainQss 𝑛 ↔ (dom ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) / ≀ (𝑟 ⋉ (◡ E ↾ 𝑛))) = 𝑛)
8	7	anbi2i 631	. . . . . 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 277	. . . . 5 ⊢ ( ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛 ↔ ( ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ EqvRels ∧ (dom ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) / ≀ (𝑟 ⋉ (◡ E ↾ 𝑛))) = 𝑛))
10	9	opabbii 5157	. . . 4 ⊢ {⟨𝑟, 𝑛⟩ ∣ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛} = {⟨𝑟, 𝑛⟩ ∣ ( ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ EqvRels ∧ (dom ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) / ≀ (𝑟 ⋉ (◡ E ↾ 𝑛))) = 𝑛)}
11		inopab 5791	. . . 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 2778	. . 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 4160	. 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 5791	. . 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 5642	. . . 4 ⊢ ( Rels × CoMembErs ) = {⟨𝑟, 𝑛⟩ ∣ (𝑟 ∈ Rels ∧ 𝑛 ∈ CoMembErs )}
16	15	ineq1i 4159	. . 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 39406	. . 3 ⊢ PetErs = {⟨𝑟, 𝑛⟩ ∣ ((𝑟 ∈ Rels ∧ 𝑛 ∈ CoMembErs ) ∧ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛)}
18	14, 16, 17	3eqtr4ri 2786	. 2 ⊢ PetErs = (( Rels × CoMembErs ) ∩ {⟨𝑟, 𝑛⟩ ∣ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛})
19		inass 4170	. 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 2785	1 ⊢ PetErs = ((( Rels × CoMembErs ) ∩ {⟨𝑟, 𝑛⟩ ∣ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ EqvRels }) ∩ {⟨𝑟, 𝑛⟩ ∣ (dom ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) / ≀ (𝑟 ⋉ (◡ E ↾ 𝑛))) = 𝑛})

Syntax hints:   ↔ wb 208   ∧ wa 398   = wceq 1550   ∈ wcel 2132  Vcvv 3444   ∩ cin 3894   class class class wbr 5090  {copab 5152   E cep 5535   × cxp 5634  ^◡ccnv 5635  dom cdm 5636   ↾ cres 5638   / cqs 8661   ⋉ cxrn 38611   ≀ ccoss 38620   Rels crels 38622   EqvRels ceqvrels 38636   DomainQss cdmqss 38643   Ers cers 38645   PetErs cpeters 38647   CoMembErs ccomembers 38649

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-rep 5217  ax-sep 5236  ax-nul 5246  ax-pow 5312  ax-pr 5380  ax-un 7703

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ne 2948  df-ral 3067  df-rex 3077  df-rab 3405  df-v 3446  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-nul 4277  df-if 4471  df-pw 4547  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4856  df-iun 4941  df-br 5091  df-opab 5153  df-mpt 5172  df-id 5531  df-eprel 5536  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-rn 5647  df-res 5648  df-ima 5649  df-iota 6462  df-fun 6508  df-fn 6509  df-f 6510  df-fo 6512  df-fv 6514  df-1st 7955  df-2nd 7956  df-ec 8664  df-qs 8668  df-xrn 38817  df-coss 38938  df-dmqss 39159  df-ers 39185  df-peters 39406

This theorem is referenced by: (None)