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Theorem sprbisymrel 45777
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) ∣ ∀𝑥𝑉𝑦𝑉 (𝑥𝑟𝑦𝑦𝑟𝑥)}
Assertion
Ref Expression
sprbisymrel (𝑉𝑊 → ∃𝑓 𝑓:𝑃1-1-onto𝑅)
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.
StepHypRef Expression
1 sprsymrelf.p . . . 4 𝑃 = 𝒫 (Pairs‘𝑉)
2 fvex 6856 . . . . 5 (Pairs‘𝑉) ∈ V
32pwex 5336 . . . 4 𝒫 (Pairs‘𝑉) ∈ V
41, 3eqeltri 2830 . . 3 𝑃 ∈ V
5 mptexg 7172 . . 3 (𝑃 ∈ V → (𝑝𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}) ∈ V)
64, 5mp1i 13 . 2 (𝑉𝑊 → (𝑝𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}) ∈ V)
7 sprsymrelf.r . . 3 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥𝑉𝑦𝑉 (𝑥𝑟𝑦𝑦𝑟𝑥)}
8 eqid 2733 . . 3 (𝑝𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}) = (𝑝𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}})
91, 7, 8sprsymrelf1o 45776 . 2 (𝑉𝑊 → (𝑝𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}):𝑃1-1-onto𝑅)
10 f1oeq1 6773 . . 3 (𝑓 = (𝑝𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}) → (𝑓:𝑃1-1-onto𝑅 ↔ (𝑝𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}):𝑃1-1-onto𝑅))
1110spcegv 3555 . 2 ((𝑝𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}) ∈ V → ((𝑝𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}):𝑃1-1-onto𝑅 → ∃𝑓 𝑓:𝑃1-1-onto𝑅))
126, 9, 11sylc 65 1 (𝑉𝑊 → ∃𝑓 𝑓:𝑃1-1-onto𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1542  wex 1782  wcel 2107  wral 3061  wrex 3070  {crab 3406  Vcvv 3444  𝒫 cpw 4561  {cpr 4589   class class class wbr 5106  {copab 5168  cmpt 5189   × cxp 5632  1-1-ontowf1o 6496  cfv 6497  Pairscspr 45755
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-spr 45756
This theorem is referenced by:  sprsymrelen  45778
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