Theorem dfpeters2 39177

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 39175 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 38955	. . . . . . 7 ⊢ (𝑛 ∈ V → ( ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛 ↔ ( ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ EqvRels ∧ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) DomainQss 𝑛)))
2	1	elv 3446	. . . . . 6 ⊢ ( ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛 ↔ ( ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ EqvRels ∧ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) DomainQss 𝑛))
3		1cossxrncnvepresex 38715	. . . . . . . . . 10 ⊢ ((𝑛 ∈ V ∧ 𝑟 ∈ V) → ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ V)
4	3	el2v 3448	. . . . . . . . 9 ⊢ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ V
5		brdmqss 38933	. . . . . . . . 9 ⊢ ((𝑛 ∈ V ∧ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ V) → ( ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) DomainQss 𝑛 ↔ (dom ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) / ≀ (𝑟 ⋉ (◡ E ↾ 𝑛))) = 𝑛))
6	4, 5	mpan2 692	. . . . . . . 8 ⊢ (𝑛 ∈ V → ( ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) DomainQss 𝑛 ↔ (dom ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) / ≀ (𝑟 ⋉ (◡ E ↾ 𝑛))) = 𝑛))
7	6	elv 3446	. . . . . . 7 ⊢ ( ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) DomainQss 𝑛 ↔ (dom ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) / ≀ (𝑟 ⋉ (◡ E ↾ 𝑛))) = 𝑛)
8	7	anbi2i 624	. . . . . 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 275	. . . . 5 ⊢ ( ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛 ↔ ( ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ EqvRels ∧ (dom ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) / ≀ (𝑟 ⋉ (◡ E ↾ 𝑛))) = 𝑛))
10	9	opabbii 5166	. . . 4 ⊢ {⟨𝑟, 𝑛⟩ ∣ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛} = {⟨𝑟, 𝑛⟩ ∣ ( ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ EqvRels ∧ (dom ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) / ≀ (𝑟 ⋉ (◡ E ↾ 𝑛))) = 𝑛)}
11		inopab 5779	. . . 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 2763	. . 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 4170	. 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 5779	. . 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 5631	. . . 4 ⊢ ( Rels × CoMembErs ) = {⟨𝑟, 𝑛⟩ ∣ (𝑟 ∈ Rels ∧ 𝑛 ∈ CoMembErs )}
16	15	ineq1i 4169	. . 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 39172	. . 3 ⊢ PetErs = {⟨𝑟, 𝑛⟩ ∣ ((𝑟 ∈ Rels ∧ 𝑛 ∈ CoMembErs ) ∧ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛)}
18	14, 16, 17	3eqtr4ri 2771	. 2 ⊢ PetErs = (( Rels × CoMembErs ) ∩ {⟨𝑟, 𝑛⟩ ∣ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛})
19		inass 4181	. 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 2770	1 ⊢ PetErs = ((( Rels × CoMembErs ) ∩ {⟨𝑟, 𝑛⟩ ∣ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ EqvRels }) ∩ {⟨𝑟, 𝑛⟩ ∣ (dom ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) / ≀ (𝑟 ⋉ (◡ E ↾ 𝑛))) = 𝑛})

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   ≀ ccoss 38386   Rels crels 38388   EqvRels ceqvrels 38402   DomainQss cdmqss 38409   Ers cers 38411   PetErs cpeters 38413   CoMembErs ccomembers 38415

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-coss 38704  df-dmqss 38925  df-ers 38951  df-peters 39172

This theorem is referenced by: (None)