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Theorem sprsymrelf 46864
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 46859 . . . . 5 (𝑝 ⊆ (Pairs‘𝑉) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉))
3 prcom 4741 . . . . . . . . . 10 {𝑥, 𝑦} = {𝑦, 𝑥}
43a1i 11 . . . . . . . . 9 (((𝑝 ⊆ (Pairs‘𝑉) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑐𝑝) → {𝑥, 𝑦} = {𝑦, 𝑥})
54eqeq2d 2739 . . . . . . . 8 (((𝑝 ⊆ (Pairs‘𝑉) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑐𝑝) → (𝑐 = {𝑥, 𝑦} ↔ 𝑐 = {𝑦, 𝑥}))
65rexbidva 3174 . . . . . . 7 ((𝑝 ⊆ (Pairs‘𝑉) ∧ (𝑥𝑉𝑦𝑉)) → (∃𝑐𝑝 𝑐 = {𝑥, 𝑦} ↔ ∃𝑐𝑝 𝑐 = {𝑦, 𝑥}))
7 df-br 5153 . . . . . . . 8 (𝑥{⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}})
8 opabidw 5530 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}} ↔ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦})
97, 8bitri 274 . . . . . . 7 (𝑥{⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}𝑦 ↔ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦})
10 vex 3477 . . . . . . . 8 𝑦 ∈ V
11 vex 3477 . . . . . . . 8 𝑥 ∈ V
12 preq12 4744 . . . . . . . . . 10 ((𝑎 = 𝑦𝑏 = 𝑥) → {𝑎, 𝑏} = {𝑦, 𝑥})
1312eqeq2d 2739 . . . . . . . . 9 ((𝑎 = 𝑦𝑏 = 𝑥) → (𝑐 = {𝑎, 𝑏} ↔ 𝑐 = {𝑦, 𝑥}))
1413rexbidv 3176 . . . . . . . 8 ((𝑎 = 𝑦𝑏 = 𝑥) → (∃𝑐𝑝 𝑐 = {𝑎, 𝑏} ↔ ∃𝑐𝑝 𝑐 = {𝑦, 𝑥}))
15 preq12 4744 . . . . . . . . . . 11 ((𝑥 = 𝑎𝑦 = 𝑏) → {𝑥, 𝑦} = {𝑎, 𝑏})
1615eqeq2d 2739 . . . . . . . . . 10 ((𝑥 = 𝑎𝑦 = 𝑏) → (𝑐 = {𝑥, 𝑦} ↔ 𝑐 = {𝑎, 𝑏}))
1716rexbidv 3176 . . . . . . . . 9 ((𝑥 = 𝑎𝑦 = 𝑏) → (∃𝑐𝑝 𝑐 = {𝑥, 𝑦} ↔ ∃𝑐𝑝 𝑐 = {𝑎, 𝑏}))
1817cbvopabv 5225 . . . . . . . 8 {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}} = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑎, 𝑏}}
1910, 11, 14, 18braba 5543 . . . . . . 7 (𝑦{⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}𝑥 ↔ ∃𝑐𝑝 𝑐 = {𝑦, 𝑥})
206, 9, 193bitr4g 313 . . . . . 6 ((𝑝 ⊆ (Pairs‘𝑉) ∧ (𝑥𝑉𝑦𝑉)) → (𝑥{⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}𝑦𝑦{⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}𝑥))
2120ralrimivva 3198 . . . . 5 (𝑝 ⊆ (Pairs‘𝑉) → ∀𝑥𝑉𝑦𝑉 (𝑥{⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}𝑦𝑦{⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}𝑥))
222, 21jca 510 . . . 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 2821 . . . . 5 (𝑝𝑃𝑝 ∈ 𝒫 (Pairs‘𝑉))
25 vex 3477 . . . . . 6 𝑝 ∈ V
2625elpw 4610 . . . . 5 (𝑝 ∈ 𝒫 (Pairs‘𝑉) ↔ 𝑝 ⊆ (Pairs‘𝑉))
2724, 26bitri 274 . . . 4 (𝑝𝑃𝑝 ⊆ (Pairs‘𝑉))
28 nfopab1 5222 . . . . . . 7 𝑥{⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}
2928nfeq2 2917 . . . . . 6 𝑥 𝑟 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}
30 nfopab2 5223 . . . . . . . 8 𝑦{⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}
3130nfeq2 2917 . . . . . . 7 𝑦 𝑟 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}
32 breq 5154 . . . . . . . 8 (𝑟 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}} → (𝑥𝑟𝑦𝑥{⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}𝑦))
33 breq 5154 . . . . . . . 8 (𝑟 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}} → (𝑦𝑟𝑥𝑦{⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}𝑥))
3432, 33bibi12d 344 . . . . . . 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 3268 . . . . . 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 3268 . . . . 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 291 . . 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 2840 . 2 (𝑝𝑃 → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}} ∈ 𝑅)
411, 40fmpti 7127 1 𝐹:𝑃𝑅
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394   = wceq 1533  wcel 2098  wral 3058  wrex 3067  {crab 3430  wss 3949  𝒫 cpw 4606  {cpr 4634  cop 4638   class class class wbr 5152  {copab 5214  cmpt 5235   × cxp 5680  wf 6549  cfv 6553  Pairscspr 46846
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 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
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 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fv 6561  df-spr 46847
This theorem is referenced by:  sprsymrelf1  46865  sprsymrelfo  46866
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