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Theorem sprvalpw 47467
Description: The set of all unordered pairs over a given set 𝑉, expressed by a restricted class abstraction. (Contributed by AV, 21-Nov-2021.)
Assertion
Ref Expression
sprvalpw (𝑉𝑊 → (Pairs‘𝑉) = {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}})
Distinct variable groups:   𝑉,𝑎,𝑏,𝑝   𝑊,𝑎,𝑏,𝑝

Proof of Theorem sprvalpw
StepHypRef Expression
1 sprval 47466 . 2 (𝑉𝑊 → (Pairs‘𝑉) = {𝑝 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}})
2 prssi 4821 . . . . . . . 8 ((𝑎𝑉𝑏𝑉) → {𝑎, 𝑏} ⊆ 𝑉)
3 eleq1 2829 . . . . . . . . 9 (𝑝 = {𝑎, 𝑏} → (𝑝 ∈ 𝒫 𝑉 ↔ {𝑎, 𝑏} ∈ 𝒫 𝑉))
4 prex 5437 . . . . . . . . . 10 {𝑎, 𝑏} ∈ V
54elpw 4604 . . . . . . . . 9 ({𝑎, 𝑏} ∈ 𝒫 𝑉 ↔ {𝑎, 𝑏} ⊆ 𝑉)
63, 5bitrdi 287 . . . . . . . 8 (𝑝 = {𝑎, 𝑏} → (𝑝 ∈ 𝒫 𝑉 ↔ {𝑎, 𝑏} ⊆ 𝑉))
72, 6syl5ibrcom 247 . . . . . . 7 ((𝑎𝑉𝑏𝑉) → (𝑝 = {𝑎, 𝑏} → 𝑝 ∈ 𝒫 𝑉))
87rexlimivv 3201 . . . . . 6 (∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏} → 𝑝 ∈ 𝒫 𝑉)
98pm4.71ri 560 . . . . 5 (∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏} ↔ (𝑝 ∈ 𝒫 𝑉 ∧ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}))
109a1i 11 . . . 4 (𝑉𝑊 → (∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏} ↔ (𝑝 ∈ 𝒫 𝑉 ∧ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏})))
1110abbidv 2808 . . 3 (𝑉𝑊 → {𝑝 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}} = {𝑝 ∣ (𝑝 ∈ 𝒫 𝑉 ∧ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏})})
12 df-rab 3437 . . 3 {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}} = {𝑝 ∣ (𝑝 ∈ 𝒫 𝑉 ∧ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏})}
1311, 12eqtr4di 2795 . 2 (𝑉𝑊 → {𝑝 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}} = {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}})
141, 13eqtrd 2777 1 (𝑉𝑊 → (Pairs‘𝑉) = {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  {cab 2714  wrex 3070  {crab 3436  wss 3951  𝒫 cpw 4600  {cpr 4628  cfv 6561  Pairscspr 47464
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-spr 47465
This theorem is referenced by:  sprvalpwn0  47470  sprel  47471  prelspr  47473
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