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Theorem sprsymrelf 42313
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 42308 . . . . 5 (𝑝 ⊆ (Pairs‘𝑉) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉))
3 prcom 4458 . . . . . . . . . 10 {𝑥, 𝑦} = {𝑦, 𝑥}
43a1i 11 . . . . . . . . 9 (((𝑝 ⊆ (Pairs‘𝑉) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑐𝑝) → {𝑥, 𝑦} = {𝑦, 𝑥})
54eqeq2d 2816 . . . . . . . 8 (((𝑝 ⊆ (Pairs‘𝑉) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑐𝑝) → (𝑐 = {𝑥, 𝑦} ↔ 𝑐 = {𝑦, 𝑥}))
65rexbidva 3237 . . . . . . 7 ((𝑝 ⊆ (Pairs‘𝑉) ∧ (𝑥𝑉𝑦𝑉)) → (∃𝑐𝑝 𝑐 = {𝑥, 𝑦} ↔ ∃𝑐𝑝 𝑐 = {𝑦, 𝑥}))
7 df-br 4845 . . . . . . . 8 (𝑥{⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}})
8 opabid 5177 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}} ↔ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦})
97, 8bitri 266 . . . . . . 7 (𝑥{⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}𝑦 ↔ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦})
10 vex 3394 . . . . . . . 8 𝑦 ∈ V
11 vex 3394 . . . . . . . 8 𝑥 ∈ V
12 preq12 4461 . . . . . . . . . 10 ((𝑎 = 𝑦𝑏 = 𝑥) → {𝑎, 𝑏} = {𝑦, 𝑥})
1312eqeq2d 2816 . . . . . . . . 9 ((𝑎 = 𝑦𝑏 = 𝑥) → (𝑐 = {𝑎, 𝑏} ↔ 𝑐 = {𝑦, 𝑥}))
1413rexbidv 3240 . . . . . . . 8 ((𝑎 = 𝑦𝑏 = 𝑥) → (∃𝑐𝑝 𝑐 = {𝑎, 𝑏} ↔ ∃𝑐𝑝 𝑐 = {𝑦, 𝑥}))
15 preq12 4461 . . . . . . . . . . 11 ((𝑥 = 𝑎𝑦 = 𝑏) → {𝑥, 𝑦} = {𝑎, 𝑏})
1615eqeq2d 2816 . . . . . . . . . 10 ((𝑥 = 𝑎𝑦 = 𝑏) → (𝑐 = {𝑥, 𝑦} ↔ 𝑐 = {𝑎, 𝑏}))
1716rexbidv 3240 . . . . . . . . 9 ((𝑥 = 𝑎𝑦 = 𝑏) → (∃𝑐𝑝 𝑐 = {𝑥, 𝑦} ↔ ∃𝑐𝑝 𝑐 = {𝑎, 𝑏}))
1817cbvopabv 4916 . . . . . . . 8 {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}} = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑎, 𝑏}}
1910, 11, 14, 18braba 5187 . . . . . . 7 (𝑦{⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}𝑥 ↔ ∃𝑐𝑝 𝑐 = {𝑦, 𝑥})
206, 9, 193bitr4g 305 . . . . . 6 ((𝑝 ⊆ (Pairs‘𝑉) ∧ (𝑥𝑉𝑦𝑉)) → (𝑥{⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}𝑦𝑦{⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}𝑥))
2120ralrimivva 3159 . . . . 5 (𝑝 ⊆ (Pairs‘𝑉) → ∀𝑥𝑉𝑦𝑉 (𝑥{⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}𝑦𝑦{⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}𝑥))
222, 21jca 503 . . . 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 2877 . . . . 5 (𝑝𝑃𝑝 ∈ 𝒫 (Pairs‘𝑉))
25 vex 3394 . . . . . 6 𝑝 ∈ V
2625elpw 4357 . . . . 5 (𝑝 ∈ 𝒫 (Pairs‘𝑉) ↔ 𝑝 ⊆ (Pairs‘𝑉))
2724, 26bitri 266 . . . 4 (𝑝𝑃𝑝 ⊆ (Pairs‘𝑉))
28 nfopab1 4913 . . . . . . 7 𝑥{⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}
2928nfeq2 2964 . . . . . 6 𝑥 𝑟 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}
30 nfopab2 4914 . . . . . . . 8 𝑦{⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}
3130nfeq2 2964 . . . . . . 7 𝑦 𝑟 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}
32 breq 4846 . . . . . . . 8 (𝑟 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}} → (𝑥𝑟𝑦𝑥{⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}𝑦))
33 breq 4846 . . . . . . . 8 (𝑟 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}} → (𝑦𝑟𝑥𝑦{⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}𝑥))
3432, 33bibi12d 336 . . . . . . 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 3171 . . . . . 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 3171 . . . . 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 3559 . . . 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 283 . . 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, 39syl6eleqr 2896 . 2 (𝑝𝑃 → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}} ∈ 𝑅)
411, 40fmpti 6604 1 𝐹:𝑃𝑅
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384   = wceq 1637  wcel 2156  wral 3096  wrex 3097  {crab 3100  wss 3769  𝒫 cpw 4351  {cpr 4372  cop 4376   class class class wbr 4844  {copab 4906  cmpt 4923   × cxp 5309  wf 6097  cfv 6101  Pairscspr 42295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-spr 42296
This theorem is referenced by:  sprsymrelf1  42314  sprsymrelfo  42315
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