Theorem sprsymrelf 48546

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 48541	. . . . 5 ⊢ (𝑝 ⊆ (Pairs‘𝑉) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉))
3		prcom 4693	. . . . . . . . . 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 2772	. . . . . . . 8 ⊢ (((𝑝 ⊆ (Pairs‘𝑉) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑝) → (𝑐 = {𝑥, 𝑦} ↔ 𝑐 = {𝑦, 𝑥}))
6	5	rexbidva 3185	. . . . . . 7 ⊢ ((𝑝 ⊆ (Pairs‘𝑉) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦} ↔ ∃𝑐 ∈ 𝑝 𝑐 = {𝑦, 𝑥}))
7		df-br 5104	. . . . . . . 8 ⊢ (𝑥{⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}})
8		opabidw 5498	. . . . . . . 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 3455	. . . . . . . 8 ⊢ 𝑦 ∈ V
11		vex 3455	. . . . . . . 8 ⊢ 𝑥 ∈ V
12		preq12 4696	. . . . . . . . . 10 ⊢ ((𝑎 = 𝑦 ∧ 𝑏 = 𝑥) → {𝑎, 𝑏} = {𝑦, 𝑥})
13	12	eqeq2d 2772	. . . . . . . . 9 ⊢ ((𝑎 = 𝑦 ∧ 𝑏 = 𝑥) → (𝑐 = {𝑎, 𝑏} ↔ 𝑐 = {𝑦, 𝑥}))
14	13	rexbidv 3187	. . . . . . . 8 ⊢ ((𝑎 = 𝑦 ∧ 𝑏 = 𝑥) → (∃𝑐 ∈ 𝑝 𝑐 = {𝑎, 𝑏} ↔ ∃𝑐 ∈ 𝑝 𝑐 = {𝑦, 𝑥}))
15		preq12 4696	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → {𝑥, 𝑦} = {𝑎, 𝑏})
16	15	eqeq2d 2772	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (𝑐 = {𝑥, 𝑦} ↔ 𝑐 = {𝑎, 𝑏}))
17	16	rexbidv 3187	. . . . . . . . 9 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦} ↔ ∃𝑐 ∈ 𝑝 𝑐 = {𝑎, 𝑏}))
18	17	cbvopabv 5178	. . . . . . . 8 ⊢ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}} = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑎, 𝑏}}
19	10, 11, 14, 18	braba 5511	. . . . . . 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 3206	. . . . 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 2853	. . . . 5 ⊢ (𝑝 ∈ 𝑃 ↔ 𝑝 ∈ 𝒫 (Pairs‘𝑉))
25		vex 3455	. . . . . 6 ⊢ 𝑝 ∈ V
26	25	elpw 4561	. . . . 5 ⊢ (𝑝 ∈ 𝒫 (Pairs‘𝑉) ↔ 𝑝 ⊆ (Pairs‘𝑉))
27	24, 26	bitri 278	. . . 4 ⊢ (𝑝 ∈ 𝑃 ↔ 𝑝 ⊆ (Pairs‘𝑉))
28		nfopab1 5175	. . . . . . 7 ⊢ Ⅎ𝑥{⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}
29	28	nfeq2 2940	. . . . . 6 ⊢ Ⅎ𝑥 𝑟 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}
30		nfopab2 5176	. . . . . . . 8 ⊢ Ⅎ𝑦{⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}
31	30	nfeq2 2940	. . . . . . 7 ⊢ Ⅎ𝑦 𝑟 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}
32		breq 5105	. . . . . . . 8 ⊢ (𝑟 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}} → (𝑥𝑟𝑦 ↔ 𝑥{⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}𝑦))
33		breq 5105	. . . . . . . 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 3276	. . . . . 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 3276	. . . . 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 3645	. . . 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 2872	. 2 ⊢ (𝑝 ∈ 𝑃 → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}} ∈ 𝑅)
41	1, 40	fmpti 7110	1 ⊢ 𝐹:𝑃⟶𝑅

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087  {crab 3413   ⊆ wss 3899  𝒫 cpw 4557  {cpr 4586  ⟨cop 4590   class class class wbr 5103  {copab 5167   ↦ cmpt 5186   × cxp 5649  ⟶wf 6533  ‘cfv 6537  Pairscspr 48528

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-spr 48529

This theorem is used by:  sprsymrelf1  48547  sprsymrelfo  48548