Theorem sprsymrelf 47855

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 47850	. . . . 5 ⊢ (𝑝 ⊆ (Pairs‘𝑉) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉))
3		prcom 4691	. . . . . . . . . 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 2748	. . . . . . . 8 ⊢ (((𝑝 ⊆ (Pairs‘𝑉) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑝) → (𝑐 = {𝑥, 𝑦} ↔ 𝑐 = {𝑦, 𝑥}))
6	5	rexbidva 3160	. . . . . . 7 ⊢ ((𝑝 ⊆ (Pairs‘𝑉) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦} ↔ ∃𝑐 ∈ 𝑝 𝑐 = {𝑦, 𝑥}))
7		df-br 5101	. . . . . . . 8 ⊢ (𝑥{⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}})
8		opabidw 5480	. . . . . . . 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 3446	. . . . . . . 8 ⊢ 𝑦 ∈ V
11		vex 3446	. . . . . . . 8 ⊢ 𝑥 ∈ V
12		preq12 4694	. . . . . . . . . 10 ⊢ ((𝑎 = 𝑦 ∧ 𝑏 = 𝑥) → {𝑎, 𝑏} = {𝑦, 𝑥})
13	12	eqeq2d 2748	. . . . . . . . 9 ⊢ ((𝑎 = 𝑦 ∧ 𝑏 = 𝑥) → (𝑐 = {𝑎, 𝑏} ↔ 𝑐 = {𝑦, 𝑥}))
14	13	rexbidv 3162	. . . . . . . 8 ⊢ ((𝑎 = 𝑦 ∧ 𝑏 = 𝑥) → (∃𝑐 ∈ 𝑝 𝑐 = {𝑎, 𝑏} ↔ ∃𝑐 ∈ 𝑝 𝑐 = {𝑦, 𝑥}))
15		preq12 4694	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → {𝑥, 𝑦} = {𝑎, 𝑏})
16	15	eqeq2d 2748	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (𝑐 = {𝑥, 𝑦} ↔ 𝑐 = {𝑎, 𝑏}))
17	16	rexbidv 3162	. . . . . . . . 9 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦} ↔ ∃𝑐 ∈ 𝑝 𝑐 = {𝑎, 𝑏}))
18	17	cbvopabv 5173	. . . . . . . 8 ⊢ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}} = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑎, 𝑏}}
19	10, 11, 14, 18	braba 5493	. . . . . . 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 3181	. . . . 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 2829	. . . . 5 ⊢ (𝑝 ∈ 𝑃 ↔ 𝑝 ∈ 𝒫 (Pairs‘𝑉))
25		vex 3446	. . . . . 6 ⊢ 𝑝 ∈ V
26	25	elpw 4560	. . . . 5 ⊢ (𝑝 ∈ 𝒫 (Pairs‘𝑉) ↔ 𝑝 ⊆ (Pairs‘𝑉))
27	24, 26	bitri 275	. . . 4 ⊢ (𝑝 ∈ 𝑃 ↔ 𝑝 ⊆ (Pairs‘𝑉))
28		nfopab1 5170	. . . . . . 7 ⊢ Ⅎ𝑥{⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}
29	28	nfeq2 2917	. . . . . 6 ⊢ Ⅎ𝑥 𝑟 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}
30		nfopab2 5171	. . . . . . . 8 ⊢ Ⅎ𝑦{⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}
31	30	nfeq2 2917	. . . . . . 7 ⊢ Ⅎ𝑦 𝑟 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}
32		breq 5102	. . . . . . . 8 ⊢ (𝑟 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}} → (𝑥𝑟𝑦 ↔ 𝑥{⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}𝑦))
33		breq 5102	. . . . . . . 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 3251	. . . . . 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 3251	. . . . 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 3648	. . . 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 2848	. 2 ⊢ (𝑝 ∈ 𝑃 → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}} ∈ 𝑅)
41	1, 40	fmpti 7066	1 ⊢ 𝐹:𝑃⟶𝑅

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  {crab 3401   ⊆ wss 3903  𝒫 cpw 4556  {cpr 4584  ⟨cop 4588   class class class wbr 5100  {copab 5162   ↦ cmpt 5181   × cxp 5630  ⟶wf 6496  ‘cfv 6500  Pairscspr 47837

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 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

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-sbc 3743  df-csb 3852  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 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-spr 47838

This theorem is referenced by:  sprsymrelf1  47856  sprsymrelfo  47857