Theorem sprbisymrel 42176

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 6314	. . . . 5 ⊢ (Pairs‘𝑉) ∈ V
3	2	pwex 4953	. . . 4 ⊢ 𝒫 (Pairs‘𝑉) ∈ V
4	1, 3	eqeltri 2799	. . 3 ⊢ 𝑃 ∈ V
5		mptexg 6600	. . 3 ⊢ (𝑃 ∈ V → (𝑝 ∈ 𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}) ∈ V)
6	4, 5	mp1i 13	. 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 2724	. . 3 ⊢ (𝑝 ∈ 𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}) = (𝑝 ∈ 𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}})
9	1, 7, 8	sprsymrelf1o 42175	. 2 ⊢ (𝑉 ∈ 𝑊 → (𝑝 ∈ 𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}):𝑃–1-1-onto→𝑅)
10		f1oeq1 6240	. . 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 3398	. 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 65	1 ⊢ (𝑉 ∈ 𝑊 → ∃𝑓 𝑓:𝑃–1-1-onto→𝑅)

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1596  ∃wex 1817   ∈ wcel 2103  ∀wral 3014  ∃wrex 3015  {crab 3018  Vcvv 3304  𝒫 cpw 4266  {cpr 4287   class class class wbr 4760  {copab 4820   ↦ cmpt 4837   × cxp 5216  –1-1-onto→wf1o 6000  ‘cfv 6001  Pairscspr 42154

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-spr 42155

This theorem is referenced by:  sprsymrelen  42177