Theorem sprsymrelf 47496

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 47491	. . . . 5 ⊢ (𝑝 ⊆ (Pairs‘𝑉) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉))
3		prcom 4696	. . . . . . . . . 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 2740	. . . . . . . 8 ⊢ (((𝑝 ⊆ (Pairs‘𝑉) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑝) → (𝑐 = {𝑥, 𝑦} ↔ 𝑐 = {𝑦, 𝑥}))
6	5	rexbidva 3155	. . . . . . 7 ⊢ ((𝑝 ⊆ (Pairs‘𝑉) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦} ↔ ∃𝑐 ∈ 𝑝 𝑐 = {𝑦, 𝑥}))
7		df-br 5108	. . . . . . . 8 ⊢ (𝑥{⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}})
8		opabidw 5484	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}} ↔ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦})
9	7, 8	bitri 275	. . . . . . 7 ⊢ (𝑥{⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}𝑦 ↔ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦})
10		vex 3451	. . . . . . . 8 ⊢ 𝑦 ∈ V
11		vex 3451	. . . . . . . 8 ⊢ 𝑥 ∈ V
12		preq12 4699	. . . . . . . . . 10 ⊢ ((𝑎 = 𝑦 ∧ 𝑏 = 𝑥) → {𝑎, 𝑏} = {𝑦, 𝑥})
13	12	eqeq2d 2740	. . . . . . . . 9 ⊢ ((𝑎 = 𝑦 ∧ 𝑏 = 𝑥) → (𝑐 = {𝑎, 𝑏} ↔ 𝑐 = {𝑦, 𝑥}))
14	13	rexbidv 3157	. . . . . . . 8 ⊢ ((𝑎 = 𝑦 ∧ 𝑏 = 𝑥) → (∃𝑐 ∈ 𝑝 𝑐 = {𝑎, 𝑏} ↔ ∃𝑐 ∈ 𝑝 𝑐 = {𝑦, 𝑥}))
15		preq12 4699	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → {𝑥, 𝑦} = {𝑎, 𝑏})
16	15	eqeq2d 2740	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (𝑐 = {𝑥, 𝑦} ↔ 𝑐 = {𝑎, 𝑏}))
17	16	rexbidv 3157	. . . . . . . . 9 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦} ↔ ∃𝑐 ∈ 𝑝 𝑐 = {𝑎, 𝑏}))
18	17	cbvopabv 5180	. . . . . . . 8 ⊢ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}} = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑎, 𝑏}}
19	10, 11, 14, 18	braba 5497	. . . . . . 7 ⊢ (𝑦{⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}𝑥 ↔ ∃𝑐 ∈ 𝑝 𝑐 = {𝑦, 𝑥})
20	6, 9, 19	3bitr4g 314	. . . . . 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 3180	. . . . 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 511	. . . 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 2820	. . . . 5 ⊢ (𝑝 ∈ 𝑃 ↔ 𝑝 ∈ 𝒫 (Pairs‘𝑉))
25		vex 3451	. . . . . 6 ⊢ 𝑝 ∈ V
26	25	elpw 4567	. . . . 5 ⊢ (𝑝 ∈ 𝒫 (Pairs‘𝑉) ↔ 𝑝 ⊆ (Pairs‘𝑉))
27	24, 26	bitri 275	. . . 4 ⊢ (𝑝 ∈ 𝑃 ↔ 𝑝 ⊆ (Pairs‘𝑉))
28		nfopab1 5177	. . . . . . 7 ⊢ Ⅎ𝑥{⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}
29	28	nfeq2 2909	. . . . . 6 ⊢ Ⅎ𝑥 𝑟 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}
30		nfopab2 5178	. . . . . . . 8 ⊢ Ⅎ𝑦{⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}
31	30	nfeq2 2909	. . . . . . 7 ⊢ Ⅎ𝑦 𝑟 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}
32		breq 5109	. . . . . . . 8 ⊢ (𝑟 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}} → (𝑥𝑟𝑦 ↔ 𝑥{⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}𝑦))
33		breq 5109	. . . . . . . 8 ⊢ (𝑟 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}} → (𝑦𝑟𝑥 ↔ 𝑦{⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}𝑥))
34	32, 33	bibi12d 345	. . . . . . 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 3250	. . . . . 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 3250	. . . . 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 3659	. . . 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 292	. . 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 2839	. 2 ⊢ (𝑝 ∈ 𝑃 → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}} ∈ 𝑅)
41	1, 40	fmpti 7084	1 ⊢ 𝐹:𝑃⟶𝑅

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  {crab 3405   ⊆ wss 3914  𝒫 cpw 4563  {cpr 4591  ⟨cop 4595   class class class wbr 5107  {copab 5169   ↦ cmpt 5188   × cxp 5636  ⟶wf 6507  ‘cfv 6511  Pairscspr 47478

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-fv 6519  df-spr 47479

This theorem is referenced by:  sprsymrelf1  47497  sprsymrelfo  47498