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Theorem sprsymrelf 45677
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 45672 . . . . 5 (𝑝 ⊆ (Pairs‘𝑉) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉))
3 prcom 4693 . . . . . . . . . 10 {𝑥, 𝑦} = {𝑦, 𝑥}
43a1i 11 . . . . . . . . 9 (((𝑝 ⊆ (Pairs‘𝑉) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑐𝑝) → {𝑥, 𝑦} = {𝑦, 𝑥})
54eqeq2d 2747 . . . . . . . 8 (((𝑝 ⊆ (Pairs‘𝑉) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑐𝑝) → (𝑐 = {𝑥, 𝑦} ↔ 𝑐 = {𝑦, 𝑥}))
65rexbidva 3173 . . . . . . 7 ((𝑝 ⊆ (Pairs‘𝑉) ∧ (𝑥𝑉𝑦𝑉)) → (∃𝑐𝑝 𝑐 = {𝑥, 𝑦} ↔ ∃𝑐𝑝 𝑐 = {𝑦, 𝑥}))
7 df-br 5106 . . . . . . . 8 (𝑥{⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}})
8 opabidw 5481 . . . . . . . 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 3449 . . . . . . . 8 𝑦 ∈ V
11 vex 3449 . . . . . . . 8 𝑥 ∈ V
12 preq12 4696 . . . . . . . . . 10 ((𝑎 = 𝑦𝑏 = 𝑥) → {𝑎, 𝑏} = {𝑦, 𝑥})
1312eqeq2d 2747 . . . . . . . . 9 ((𝑎 = 𝑦𝑏 = 𝑥) → (𝑐 = {𝑎, 𝑏} ↔ 𝑐 = {𝑦, 𝑥}))
1413rexbidv 3175 . . . . . . . 8 ((𝑎 = 𝑦𝑏 = 𝑥) → (∃𝑐𝑝 𝑐 = {𝑎, 𝑏} ↔ ∃𝑐𝑝 𝑐 = {𝑦, 𝑥}))
15 preq12 4696 . . . . . . . . . . 11 ((𝑥 = 𝑎𝑦 = 𝑏) → {𝑥, 𝑦} = {𝑎, 𝑏})
1615eqeq2d 2747 . . . . . . . . . 10 ((𝑥 = 𝑎𝑦 = 𝑏) → (𝑐 = {𝑥, 𝑦} ↔ 𝑐 = {𝑎, 𝑏}))
1716rexbidv 3175 . . . . . . . . 9 ((𝑥 = 𝑎𝑦 = 𝑏) → (∃𝑐𝑝 𝑐 = {𝑥, 𝑦} ↔ ∃𝑐𝑝 𝑐 = {𝑎, 𝑏}))
1817cbvopabv 5178 . . . . . . . 8 {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}} = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑎, 𝑏}}
1910, 11, 14, 18braba 5494 . . . . . . 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 3197 . . . . 5 (𝑝 ⊆ (Pairs‘𝑉) → ∀𝑥𝑉𝑦𝑉 (𝑥{⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}𝑦𝑦{⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}𝑥))
222, 21jca 512 . . . 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 2829 . . . . 5 (𝑝𝑃𝑝 ∈ 𝒫 (Pairs‘𝑉))
25 vex 3449 . . . . . 6 𝑝 ∈ V
2625elpw 4564 . . . . 5 (𝑝 ∈ 𝒫 (Pairs‘𝑉) ↔ 𝑝 ⊆ (Pairs‘𝑉))
2724, 26bitri 274 . . . 4 (𝑝𝑃𝑝 ⊆ (Pairs‘𝑉))
28 nfopab1 5175 . . . . . . 7 𝑥{⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}
2928nfeq2 2924 . . . . . 6 𝑥 𝑟 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}
30 nfopab2 5176 . . . . . . . 8 𝑦{⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}
3130nfeq2 2924 . . . . . . 7 𝑦 𝑟 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}
32 breq 5107 . . . . . . . 8 (𝑟 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}} → (𝑥𝑟𝑦𝑥{⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}𝑦))
33 breq 5107 . . . . . . . 8 (𝑟 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}} → (𝑦𝑟𝑥𝑦{⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}𝑥))
3432, 33bibi12d 345 . . . . . . 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 3256 . . . . . 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 3256 . . . . 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 3645 . . . 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 2849 . 2 (𝑝𝑃 → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}} ∈ 𝑅)
411, 40fmpti 7060 1 𝐹:𝑃𝑅
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3064  wrex 3073  {crab 3407  wss 3910  𝒫 cpw 4560  {cpr 4588  cop 4592   class class class wbr 5105  {copab 5167  cmpt 5188   × cxp 5631  wf 6492  cfv 6496  Pairscspr 45659
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fv 6504  df-spr 45660
This theorem is referenced by:  sprsymrelf1  45678  sprsymrelfo  45679
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