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Theorem sprvalpw 42255
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 42254 . 2 (𝑉𝑊 → (Pairs‘𝑉) = {𝑝 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}})
2 prssi 4488 . . . . . . . 8 ((𝑎𝑉𝑏𝑉) → {𝑎, 𝑏} ⊆ 𝑉)
3 eleq1 2838 . . . . . . . . 9 (𝑝 = {𝑎, 𝑏} → (𝑝 ∈ 𝒫 𝑉 ↔ {𝑎, 𝑏} ∈ 𝒫 𝑉))
4 prex 5038 . . . . . . . . . 10 {𝑎, 𝑏} ∈ V
54elpw 4304 . . . . . . . . 9 ({𝑎, 𝑏} ∈ 𝒫 𝑉 ↔ {𝑎, 𝑏} ⊆ 𝑉)
63, 5syl6bb 276 . . . . . . . 8 (𝑝 = {𝑎, 𝑏} → (𝑝 ∈ 𝒫 𝑉 ↔ {𝑎, 𝑏} ⊆ 𝑉))
72, 6syl5ibrcom 237 . . . . . . 7 ((𝑎𝑉𝑏𝑉) → (𝑝 = {𝑎, 𝑏} → 𝑝 ∈ 𝒫 𝑉))
87rexlimivv 3184 . . . . . 6 (∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏} → 𝑝 ∈ 𝒫 𝑉)
98pm4.71ri 550 . . . . 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 2890 . . 3 (𝑉𝑊 → {𝑝 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}} = {𝑝 ∣ (𝑝 ∈ 𝒫 𝑉 ∧ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏})})
12 df-rab 3070 . . 3 {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}} = {𝑝 ∣ (𝑝 ∈ 𝒫 𝑉 ∧ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏})}
1311, 12syl6eqr 2823 . 2 (𝑉𝑊 → {𝑝 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}} = {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}})
141, 13eqtrd 2805 1 (𝑉𝑊 → (Pairs‘𝑉) = {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  {cab 2757  wrex 3062  {crab 3065  wss 3723  𝒫 cpw 4298  {cpr 4319  cfv 6030  Pairscspr 42252
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-spr 42253
This theorem is referenced by:  sprvalpwn0  42258  sprel  42259  prelspr  42261
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