Theorem dfpeters2 39254

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 39252 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 39032	. . . . . . 7 ⊢ (𝑛 ∈ V → ( ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛 ↔ ( ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ EqvRels ∧ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) DomainQss 𝑛)))
2	1	elv 3447	. . . . . 6 ⊢ ( ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛 ↔ ( ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ EqvRels ∧ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) DomainQss 𝑛))
3		1cossxrncnvepresex 38792	. . . . . . . . . 10 ⊢ ((𝑛 ∈ V ∧ 𝑟 ∈ V) → ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ V)
4	3	el2v 3449	. . . . . . . . 9 ⊢ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ V
5		brdmqss 39010	. . . . . . . . 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 3447	. . . . . . 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 5167	. . . 4 ⊢ {⟨𝑟, 𝑛⟩ ∣ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛} = {⟨𝑟, 𝑛⟩ ∣ ( ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ EqvRels ∧ (dom ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) / ≀ (𝑟 ⋉ (◡ E ↾ 𝑛))) = 𝑛)}
11		inopab 5788	. . . 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 4171	. 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 5788	. . 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 5640	. . . 4 ⊢ ( Rels × CoMembErs ) = {⟨𝑟, 𝑛⟩ ∣ (𝑟 ∈ Rels ∧ 𝑛 ∈ CoMembErs )}
16	15	ineq1i 4170	. . 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 39249	. . 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 4182	. 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 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   ≀ ccoss 38463   Rels crels 38465   EqvRels ceqvrels 38479   DomainQss cdmqss 38486   Ers cers 38488   PetErs cpeters 38490   CoMembErs ccomembers 38492

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-coss 38781  df-dmqss 39002  df-ers 39028  df-peters 39249

This theorem is referenced by: (None)