Theorem sprbisymrel 48525

Description: There is a bijection between the subsets of the set of pairs over a fixed set 𝑉 and the symmetric relations 𝑅 on the fixed set 𝑉. (Contributed by AV, 23-Nov-2021.)

Hypotheses

Ref	Expression
sprsymrelf.p	⊢ 𝑃 = 𝒫 (Pairs‘𝑉)
sprsymrelf.r	⊢ 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)}

Assertion

Ref	Expression
sprbisymrel	⊢ (𝑉 ∈ 𝑊 → ∃𝑓 𝑓:𝑃–1-1-onto→𝑅)

Distinct variable groups:   𝑥,𝑉,𝑦,𝑟   𝑥,𝑓,𝑦   𝑃,𝑓   𝑅,𝑓   𝑉,𝑟   𝑥,𝑊,𝑦

Allowed substitution hints:   𝑃(𝑥, 𝑦, 𝑟)   𝑅(𝑥, 𝑦, 𝑟)   𝑉(𝑓)   𝑊(𝑓, 𝑟)

Proof of Theorem sprbisymrel

Dummy variables 𝑝 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sprsymrelf.p	. . . 4 ⊢ 𝑃 = 𝒫 (Pairs‘𝑉)
2		fvex 6890	. . . . 5 ⊢ (Pairs‘𝑉) ∈ V
3	2	pwex 5342	. . . 4 ⊢ 𝒫 (Pairs‘𝑉) ∈ V
4	1, 3	eqeltri 2857	. . 3 ⊢ 𝑃 ∈ V
5		mptexg 7219	. . 3 ⊢ (𝑃 ∈ V → (𝑝 ∈ 𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}) ∈ V)
6	4, 5	mp1i 14	. 2 ⊢ (𝑉 ∈ 𝑊 → (𝑝 ∈ 𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}) ∈ V)
7		sprsymrelf.r	. . 3 ⊢ 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)}
8		eqid 2761	. . 3 ⊢ (𝑝 ∈ 𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}) = (𝑝 ∈ 𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}})
9	1, 7, 8	sprsymrelf1o 48524	. 2 ⊢ (𝑉 ∈ 𝑊 → (𝑝 ∈ 𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}):𝑃–1-1-onto→𝑅)
10		f1oeq1 6804	. . 3 ⊢ (𝑓 = (𝑝 ∈ 𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}) → (𝑓:𝑃–1-1-onto→𝑅 ↔ (𝑝 ∈ 𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}):𝑃–1-1-onto→𝑅))
11	10	spcegv 3552	. 2 ⊢ ((𝑝 ∈ 𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}) ∈ V → ((𝑝 ∈ 𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}):𝑃–1-1-onto→𝑅 → ∃𝑓 𝑓:𝑃–1-1-onto→𝑅))
12	6, 9, 11	sylc 66	1 ⊢ (𝑉 ∈ 𝑊 → ∃𝑓 𝑓:𝑃–1-1-onto→𝑅)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087  {crab 3413  Vcvv 3451  𝒫 cpw 4557  {cpr 4586   class class class wbr 5103  {copab 5167   ↦ cmpt 5186   × cxp 5649  –1-1-onto→wf1o 6530  ‘cfv 6531  Pairscspr 48503

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 7740

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-reu 3367  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 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-spr 48504

This theorem is used by:  sprsymrelen  48526