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Theorem sprbisymrel 46901
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 6904 . . . . 5 (Pairs‘𝑉) ∈ V
32pwex 5374 . . . 4 𝒫 (Pairs‘𝑉) ∈ V
41, 3eqeltri 2821 . . 3 𝑃 ∈ V
5 mptexg 7228 . . 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 2725 . . 3 (𝑝𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}) = (𝑝𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}})
91, 7, 8sprsymrelf1o 46900 . 2 (𝑉𝑊 → (𝑝𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}):𝑃1-1-onto𝑅)
10 f1oeq1 6821 . . 3 (𝑓 = (𝑝𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}) → (𝑓:𝑃1-1-onto𝑅 ↔ (𝑝𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}):𝑃1-1-onto𝑅))
1110spcegv 3577 . 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 1533  wex 1773  wcel 2098  wral 3051  wrex 3060  {crab 3419  Vcvv 3463  𝒫 cpw 4598  {cpr 4626   class class class wbr 5143  {copab 5205  cmpt 5226   × cxp 5670  1-1-ontowf1o 6541  cfv 6542  Pairscspr 46879
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-spr 46880
This theorem is referenced by:  sprsymrelen  46902
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