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Theorem sprsymrelf 45777
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 45772 . . . . 5 (𝑝 ⊆ (Pairs‘𝑉) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉))
3 prcom 4697 . . . . . . . . . 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 3170 . . . . . . 7 ((𝑝 ⊆ (Pairs‘𝑉) ∧ (𝑥𝑉𝑦𝑉)) → (∃𝑐𝑝 𝑐 = {𝑥, 𝑦} ↔ ∃𝑐𝑝 𝑐 = {𝑦, 𝑥}))
7 df-br 5110 . . . . . . . 8 (𝑥{⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}})
8 opabidw 5485 . . . . . . . 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 3451 . . . . . . . 8 𝑦 ∈ V
11 vex 3451 . . . . . . . 8 𝑥 ∈ V
12 preq12 4700 . . . . . . . . . 10 ((𝑎 = 𝑦𝑏 = 𝑥) → {𝑎, 𝑏} = {𝑦, 𝑥})
1312eqeq2d 2744 . . . . . . . . 9 ((𝑎 = 𝑦𝑏 = 𝑥) → (𝑐 = {𝑎, 𝑏} ↔ 𝑐 = {𝑦, 𝑥}))
1413rexbidv 3172 . . . . . . . 8 ((𝑎 = 𝑦𝑏 = 𝑥) → (∃𝑐𝑝 𝑐 = {𝑎, 𝑏} ↔ ∃𝑐𝑝 𝑐 = {𝑦, 𝑥}))
15 preq12 4700 . . . . . . . . . . 11 ((𝑥 = 𝑎𝑦 = 𝑏) → {𝑥, 𝑦} = {𝑎, 𝑏})
1615eqeq2d 2744 . . . . . . . . . 10 ((𝑥 = 𝑎𝑦 = 𝑏) → (𝑐 = {𝑥, 𝑦} ↔ 𝑐 = {𝑎, 𝑏}))
1716rexbidv 3172 . . . . . . . . 9 ((𝑥 = 𝑎𝑦 = 𝑏) → (∃𝑐𝑝 𝑐 = {𝑥, 𝑦} ↔ ∃𝑐𝑝 𝑐 = {𝑎, 𝑏}))
1817cbvopabv 5182 . . . . . . . 8 {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}} = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑎, 𝑏}}
1910, 11, 14, 18braba 5498 . . . . . . 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 3194 . . . . 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 3451 . . . . . 6 𝑝 ∈ V
2625elpw 4568 . . . . 5 (𝑝 ∈ 𝒫 (Pairs‘𝑉) ↔ 𝑝 ⊆ (Pairs‘𝑉))
2724, 26bitri 275 . . . 4 (𝑝𝑃𝑝 ⊆ (Pairs‘𝑉))
28 nfopab1 5179 . . . . . . 7 𝑥{⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}
2928nfeq2 2921 . . . . . 6 𝑥 𝑟 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}
30 nfopab2 5180 . . . . . . . 8 𝑦{⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}
3130nfeq2 2921 . . . . . . 7 𝑦 𝑟 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}
32 breq 5111 . . . . . . . 8 (𝑟 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}} → (𝑥𝑟𝑦𝑥{⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}}𝑦))
33 breq 5111 . . . . . . . 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 3255 . . . . . 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 3255 . . . . 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 3649 . . . 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 7064 1 𝐹:𝑃𝑅
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3061  wrex 3070  {crab 3406  wss 3914  𝒫 cpw 4564  {cpr 4592  cop 4596   class class class wbr 5109  {copab 5171  cmpt 5192   × cxp 5635  wf 6496  cfv 6500  Pairscspr 45759
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 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
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 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-spr 45760
This theorem is referenced by:  sprsymrelf1  45778  sprsymrelfo  45779
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