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Theorem sprsymrelf 46163
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) ∣ ∀𝑥𝑉𝑦𝑉 (𝑥𝑟𝑦𝑦𝑟𝑥)}
sprsymrelf.f 𝐹 = (𝑝𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}})
Assertion
Ref Expression
sprsymrelf 𝐹:𝑃𝑅
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.
StepHypRef Expression
1 sprsymrelf.f . 2 𝐹 = (𝑝𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}})
2 sprsymrelfvlem 46158 . . . . 5 (𝑝 ⊆ (Pairs‘𝑉) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉))
3 prcom 4737 . . . . . . . . . 10 {𝑥, 𝑦} = {𝑦, 𝑥}
43a1i 11 . . . . . . . . 9 (((𝑝 ⊆ (Pairs‘𝑉) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑐𝑝) → {𝑥, 𝑦} = {𝑦, 𝑥})
54eqeq2d 2744 . . . . . . . 8 (((𝑝 ⊆ (Pairs‘𝑉) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑐𝑝) → (𝑐 = {𝑥, 𝑦} ↔ 𝑐 = {𝑦, 𝑥}))
65rexbidva 3177 . . . . . . 7 ((𝑝 ⊆ (Pairs‘𝑉) ∧ (𝑥𝑉𝑦𝑉)) → (∃𝑐𝑝 𝑐 = {𝑥, 𝑦} ↔ ∃𝑐𝑝 𝑐 = {𝑦, 𝑥}))
7 df-br 5150 . . . . . . . 8 (𝑥{⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}})
8 opabidw 5525 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}} ↔ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦})
97, 8bitri 275 . . . . . . 7 (𝑥{⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}𝑦 ↔ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦})
10 vex 3479 . . . . . . . 8 𝑦 ∈ V
11 vex 3479 . . . . . . . 8 𝑥 ∈ V
12 preq12 4740 . . . . . . . . . 10 ((𝑎 = 𝑦𝑏 = 𝑥) → {𝑎, 𝑏} = {𝑦, 𝑥})
1312eqeq2d 2744 . . . . . . . . 9 ((𝑎 = 𝑦𝑏 = 𝑥) → (𝑐 = {𝑎, 𝑏} ↔ 𝑐 = {𝑦, 𝑥}))
1413rexbidv 3179 . . . . . . . 8 ((𝑎 = 𝑦𝑏 = 𝑥) → (∃𝑐𝑝 𝑐 = {𝑎, 𝑏} ↔ ∃𝑐𝑝 𝑐 = {𝑦, 𝑥}))
15 preq12 4740 . . . . . . . . . . 11 ((𝑥 = 𝑎𝑦 = 𝑏) → {𝑥, 𝑦} = {𝑎, 𝑏})
1615eqeq2d 2744 . . . . . . . . . 10 ((𝑥 = 𝑎𝑦 = 𝑏) → (𝑐 = {𝑥, 𝑦} ↔ 𝑐 = {𝑎, 𝑏}))
1716rexbidv 3179 . . . . . . . . 9 ((𝑥 = 𝑎𝑦 = 𝑏) → (∃𝑐𝑝 𝑐 = {𝑥, 𝑦} ↔ ∃𝑐𝑝 𝑐 = {𝑎, 𝑏}))
1817cbvopabv 5222 . . . . . . . 8 {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}} = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑎, 𝑏}}
1910, 11, 14, 18braba 5538 . . . . . . 7 (𝑦{⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}𝑥 ↔ ∃𝑐𝑝 𝑐 = {𝑦, 𝑥})
206, 9, 193bitr4g 314 . . . . . 6 ((𝑝 ⊆ (Pairs‘𝑉) ∧ (𝑥𝑉𝑦𝑉)) → (𝑥{⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}𝑦𝑦{⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}𝑥))
2120ralrimivva 3201 . . . . 5 (𝑝 ⊆ (Pairs‘𝑉) → ∀𝑥𝑉𝑦𝑉 (𝑥{⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}𝑦𝑦{⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}𝑥))
222, 21jca 513 . . . 4 (𝑝 ⊆ (Pairs‘𝑉) → ({⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥{⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}𝑦𝑦{⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}𝑥)))
23 sprsymrelf.p . . . . . 6 𝑃 = 𝒫 (Pairs‘𝑉)
2423eleq2i 2826 . . . . 5 (𝑝𝑃𝑝 ∈ 𝒫 (Pairs‘𝑉))
25 vex 3479 . . . . . 6 𝑝 ∈ V
2625elpw 4607 . . . . 5 (𝑝 ∈ 𝒫 (Pairs‘𝑉) ↔ 𝑝 ⊆ (Pairs‘𝑉))
2724, 26bitri 275 . . . 4 (𝑝𝑃𝑝 ⊆ (Pairs‘𝑉))
28 nfopab1 5219 . . . . . . 7 𝑥{⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}
2928nfeq2 2921 . . . . . 6 𝑥 𝑟 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}
30 nfopab2 5220 . . . . . . . 8 𝑦{⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}
3130nfeq2 2921 . . . . . . 7 𝑦 𝑟 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}
32 breq 5151 . . . . . . . 8 (𝑟 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}} → (𝑥𝑟𝑦𝑥{⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}𝑦))
33 breq 5151 . . . . . . . 8 (𝑟 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}} → (𝑦𝑟𝑥𝑦{⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}𝑥))
3432, 33bibi12d 346 . . . . . . 7 (𝑟 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}} → ((𝑥𝑟𝑦𝑦𝑟𝑥) ↔ (𝑥{⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}𝑦𝑦{⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}𝑥)))
3531, 34ralbid 3271 . . . . . 6 (𝑟 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}} → (∀𝑦𝑉 (𝑥𝑟𝑦𝑦𝑟𝑥) ↔ ∀𝑦𝑉 (𝑥{⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}𝑦𝑦{⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}𝑥)))
3629, 35ralbid 3271 . . . . 5 (𝑟 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}} → (∀𝑥𝑉𝑦𝑉 (𝑥𝑟𝑦𝑦𝑟𝑥) ↔ ∀𝑥𝑉𝑦𝑉 (𝑥{⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}𝑦𝑦{⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}𝑥)))
3736elrab 3684 . . . 4 ({⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}} ∈ {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥𝑉𝑦𝑉 (𝑥𝑟𝑦𝑦𝑟𝑥)} ↔ ({⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥{⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}𝑦𝑦{⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}𝑥)))
3822, 27, 373imtr4i 292 . . 3 (𝑝𝑃 → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}} ∈ {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥𝑉𝑦𝑉 (𝑥𝑟𝑦𝑦𝑟𝑥)})
39 sprsymrelf.r . . 3 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥𝑉𝑦𝑉 (𝑥𝑟𝑦𝑦𝑟𝑥)}
4038, 39eleqtrrdi 2845 . 2 (𝑝𝑃 → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}} ∈ 𝑅)
411, 40fmpti 7112 1 𝐹:𝑃𝑅
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3062  wrex 3071  {crab 3433  wss 3949  𝒫 cpw 4603  {cpr 4631  cop 4635   class class class wbr 5149  {copab 5211  cmpt 5232   × cxp 5675  wf 6540  cfv 6544  Pairscspr 46145
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 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
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 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-spr 46146
This theorem is referenced by:  sprsymrelf1  46164  sprsymrelfo  46165
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