Theorem sprsymrelf 48302

Description: The mapping 𝐹 is a function from the subsets of the set of pairs over a fixed set 𝑉 into the symmetric relations 𝑅 on the fixed set 𝑉. (Contributed by AV, 19-Nov-2021.)

Hypotheses

Ref	Expression
sprsymrelf.p	⊢ 𝑃 = 𝒫 (Pairs‘𝑉)
sprsymrelf.r	⊢ 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)}
sprsymrelf.f	⊢ 𝐹 = (𝑝 ∈ 𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}})

Assertion

Ref	Expression
sprsymrelf	⊢ 𝐹:𝑃⟶𝑅

Distinct variable groups:   𝑃,𝑝   𝑉,𝑐,𝑥,𝑦   𝑝,𝑐,𝑥,𝑦,𝑟   𝑅,𝑝   𝑉,𝑟,𝑐,𝑥,𝑦

Allowed substitution hints:   𝑃(𝑥, 𝑦, 𝑟, 𝑐)   𝑅(𝑥, 𝑦, 𝑟, 𝑐)   𝐹(𝑥, 𝑦, 𝑟, 𝑝, 𝑐)   𝑉(𝑝)

Proof of Theorem sprsymrelf

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sprsymrelf.f	. 2 ⊢ 𝐹 = (𝑝 ∈ 𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}})
2		sprsymrelfvlem 48297	. . . . 5 ⊢ (𝑝 ⊆ (Pairs‘𝑉) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉))
3		prcom 4700	. . . . . . . . . 10 ⊢ {𝑥, 𝑦} = {𝑦, 𝑥}
4	3	a1i 11	. . . . . . . . 9 ⊢ (((𝑝 ⊆ (Pairs‘𝑉) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑝) → {𝑥, 𝑦} = {𝑦, 𝑥})
5	4	eqeq2d 2776	. . . . . . . 8 ⊢ (((𝑝 ⊆ (Pairs‘𝑉) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑝) → (𝑐 = {𝑥, 𝑦} ↔ 𝑐 = {𝑦, 𝑥}))
6	5	rexbidva 3189	. . . . . . 7 ⊢ ((𝑝 ⊆ (Pairs‘𝑉) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦} ↔ ∃𝑐 ∈ 𝑝 𝑐 = {𝑦, 𝑥}))
7		df-br 5112	. . . . . . . 8 ⊢ (𝑥{⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}})
8		opabidw 5510	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}} ↔ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦})
9	7, 8	bitri 278	. . . . . . 7 ⊢ (𝑥{⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}𝑦 ↔ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦})
10		vex 3461	. . . . . . . 8 ⊢ 𝑦 ∈ V
11		vex 3461	. . . . . . . 8 ⊢ 𝑥 ∈ V
12		preq12 4703	. . . . . . . . . 10 ⊢ ((𝑎 = 𝑦 ∧ 𝑏 = 𝑥) → {𝑎, 𝑏} = {𝑦, 𝑥})
13	12	eqeq2d 2776	. . . . . . . . 9 ⊢ ((𝑎 = 𝑦 ∧ 𝑏 = 𝑥) → (𝑐 = {𝑎, 𝑏} ↔ 𝑐 = {𝑦, 𝑥}))
14	13	rexbidv 3191	. . . . . . . 8 ⊢ ((𝑎 = 𝑦 ∧ 𝑏 = 𝑥) → (∃𝑐 ∈ 𝑝 𝑐 = {𝑎, 𝑏} ↔ ∃𝑐 ∈ 𝑝 𝑐 = {𝑦, 𝑥}))
15		preq12 4703	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → {𝑥, 𝑦} = {𝑎, 𝑏})
16	15	eqeq2d 2776	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (𝑐 = {𝑥, 𝑦} ↔ 𝑐 = {𝑎, 𝑏}))
17	16	rexbidv 3191	. . . . . . . . 9 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦} ↔ ∃𝑐 ∈ 𝑝 𝑐 = {𝑎, 𝑏}))
18	17	cbvopabv 5186	. . . . . . . 8 ⊢ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}} = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑎, 𝑏}}
19	10, 11, 14, 18	braba 5523	. . . . . . 7 ⊢ (𝑦{⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}𝑥 ↔ ∃𝑐 ∈ 𝑝 𝑐 = {𝑦, 𝑥})
20	6, 9, 19	3bitr4g 317	. . . . . 6 ⊢ ((𝑝 ⊆ (Pairs‘𝑉) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥{⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}𝑦 ↔ 𝑦{⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}𝑥))
21	20	ralrimivva 3210	. . . . 5 ⊢ (𝑝 ⊆ (Pairs‘𝑉) → ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥{⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}𝑦 ↔ 𝑦{⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}𝑥))
22	2, 21	jca 521	. . . 4 ⊢ (𝑝 ⊆ (Pairs‘𝑉) → ({⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥{⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}𝑦 ↔ 𝑦{⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}𝑥)))
23		sprsymrelf.p	. . . . . 6 ⊢ 𝑃 = 𝒫 (Pairs‘𝑉)
24	23	eleq2i 2857	. . . . 5 ⊢ (𝑝 ∈ 𝑃 ↔ 𝑝 ∈ 𝒫 (Pairs‘𝑉))
25		vex 3461	. . . . . 6 ⊢ 𝑝 ∈ V
26	25	elpw 4568	. . . . 5 ⊢ (𝑝 ∈ 𝒫 (Pairs‘𝑉) ↔ 𝑝 ⊆ (Pairs‘𝑉))
27	24, 26	bitri 278	. . . 4 ⊢ (𝑝 ∈ 𝑃 ↔ 𝑝 ⊆ (Pairs‘𝑉))
28		nfopab1 5183	. . . . . . 7 ⊢ Ⅎ𝑥{⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}
29	28	nfeq2 2944	. . . . . 6 ⊢ Ⅎ𝑥 𝑟 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}
30		nfopab2 5184	. . . . . . . 8 ⊢ Ⅎ𝑦{⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}
31	30	nfeq2 2944	. . . . . . 7 ⊢ Ⅎ𝑦 𝑟 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}
32		breq 5113	. . . . . . . 8 ⊢ (𝑟 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}} → (𝑥𝑟𝑦 ↔ 𝑥{⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}𝑦))
33		breq 5113	. . . . . . . 8 ⊢ (𝑟 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}} → (𝑦𝑟𝑥 ↔ 𝑦{⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}𝑥))
34	32, 33	bibi12d 348	. . . . . . 7 ⊢ (𝑟 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}} → ((𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥) ↔ (𝑥{⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}𝑦 ↔ 𝑦{⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}𝑥)))
35	31, 34	ralbid 3280	. . . . . 6 ⊢ (𝑟 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}} → (∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥) ↔ ∀𝑦 ∈ 𝑉 (𝑥{⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}𝑦 ↔ 𝑦{⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}𝑥)))
36	29, 35	ralbid 3280	. . . . 5 ⊢ (𝑟 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}} → (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥) ↔ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥{⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}𝑦 ↔ 𝑦{⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}𝑥)))
37	36	elrab 3652	. . . 4 ⊢ ({⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}} ∈ {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)} ↔ ({⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥{⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}𝑦 ↔ 𝑦{⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}𝑥)))
38	22, 27, 37	3imtr4i 295	. . 3 ⊢ (𝑝 ∈ 𝑃 → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}} ∈ {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)})
39		sprsymrelf.r	. . 3 ⊢ 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)}
40	38, 39	eleqtrrdi 2876	. 2 ⊢ (𝑝 ∈ 𝑃 → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}} ∈ 𝑅)
41	1, 40	fmpti 7111	1 ⊢ 𝐹:𝑃⟶𝑅

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3081  ∃wrex 3091  {crab 3418   ⊆ wss 3906  𝒫 cpw 4564  {cpr 4593  ⟨cop 4597   class class class wbr 5111  {copab 5175   ↦ cmpt 5194   × cxp 5661  ⟶wf 6536  ‘cfv 6540  Pairscspr 48284

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

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 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-spr 48285

This theorem is used by:  sprsymrelf1  48303  sprsymrelfo  48304