Theorem dfpeters2 39286

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 39284 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 39064	. . . . . . 7 ⊢ (𝑛 ∈ V → ( ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛 ↔ ( ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ EqvRels ∧ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) DomainQss 𝑛)))
2	1	elv 3435	. . . . . 6 ⊢ ( ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛 ↔ ( ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ EqvRels ∧ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) DomainQss 𝑛))
3		1cossxrncnvepresex 38824	. . . . . . . . . 10 ⊢ ((𝑛 ∈ V ∧ 𝑟 ∈ V) → ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ V)
4	3	el2v 3437	. . . . . . . . 9 ⊢ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ V
5		brdmqss 39042	. . . . . . . . 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 3435	. . . . . . 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 5153	. . . 4 ⊢ {⟨𝑟, 𝑛⟩ ∣ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛} = {⟨𝑟, 𝑛⟩ ∣ ( ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ EqvRels ∧ (dom ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) / ≀ (𝑟 ⋉ (◡ E ↾ 𝑛))) = 𝑛)}
11		inopab 5776	. . . 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 4158	. 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 5776	. . 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 5628	. . . 4 ⊢ ( Rels × CoMembErs ) = {⟨𝑟, 𝑛⟩ ∣ (𝑟 ∈ Rels ∧ 𝑛 ∈ CoMembErs )}
16	15	ineq1i 4157	. . 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 39281	. . 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 4169	. 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 3430   ∩ cin 3889   class class class wbr 5086  {copab 5148   E cep 5521   × cxp 5620  ^◡ccnv 5621  dom cdm 5622   ↾ cres 5624   / cqs 8633   ⋉ cxrn 38486   ≀ ccoss 38495   Rels crels 38497   EqvRels ceqvrels 38511   DomainQss cdmqss 38518   Ers cers 38520   PetErs cpeters 38522   CoMembErs ccomembers 38524

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 5212  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5368  ax-un 7680

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 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5517  df-eprel 5522  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-fo 6496  df-fv 6498  df-1st 7933  df-2nd 7934  df-ec 8636  df-qs 8640  df-xrn 38692  df-coss 38813  df-dmqss 39034  df-ers 39060  df-peters 39281

This theorem is referenced by: (None)