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Theorem sprval 46657
Description: The set of all unordered pairs over a given set 𝑉. (Contributed by AV, 21-Nov-2021.)
Assertion
Ref Expression
sprval (𝑉𝑊 → (Pairs‘𝑉) = {𝑝 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}})
Distinct variable groups:   𝑉,𝑎,𝑏,𝑝   𝑊,𝑎,𝑏,𝑝

Proof of Theorem sprval
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 df-spr 46656 . . 3 Pairs = (𝑣 ∈ V ↦ {𝑝 ∣ ∃𝑎𝑣𝑏𝑣 𝑝 = {𝑎, 𝑏}})
21a1i 11 . 2 (𝑉𝑊 → Pairs = (𝑣 ∈ V ↦ {𝑝 ∣ ∃𝑎𝑣𝑏𝑣 𝑝 = {𝑎, 𝑏}}))
3 id 22 . . . . 5 (𝑣 = 𝑉𝑣 = 𝑉)
4 rexeq 3313 . . . . 5 (𝑣 = 𝑉 → (∃𝑏𝑣 𝑝 = {𝑎, 𝑏} ↔ ∃𝑏𝑉 𝑝 = {𝑎, 𝑏}))
53, 4rexeqbidv 3335 . . . 4 (𝑣 = 𝑉 → (∃𝑎𝑣𝑏𝑣 𝑝 = {𝑎, 𝑏} ↔ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}))
65adantl 481 . . 3 ((𝑉𝑊𝑣 = 𝑉) → (∃𝑎𝑣𝑏𝑣 𝑝 = {𝑎, 𝑏} ↔ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}))
76abbidv 2793 . 2 ((𝑉𝑊𝑣 = 𝑉) → {𝑝 ∣ ∃𝑎𝑣𝑏𝑣 𝑝 = {𝑎, 𝑏}} = {𝑝 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}})
8 elex 3485 . 2 (𝑉𝑊𝑉 ∈ V)
9 zfpair2 5419 . . . . . . . 8 {𝑎, 𝑏} ∈ V
10 eueq 3697 . . . . . . . 8 ({𝑎, 𝑏} ∈ V ↔ ∃!𝑝 𝑝 = {𝑎, 𝑏})
119, 10mpbi 229 . . . . . . 7 ∃!𝑝 𝑝 = {𝑎, 𝑏}
12 euabex 5452 . . . . . . 7 (∃!𝑝 𝑝 = {𝑎, 𝑏} → {𝑝𝑝 = {𝑎, 𝑏}} ∈ V)
1311, 12mp1i 13 . . . . . 6 (𝑉𝑊 → {𝑝𝑝 = {𝑎, 𝑏}} ∈ V)
1413ralrimivw 3142 . . . . 5 (𝑉𝑊 → ∀𝑏𝑉 {𝑝𝑝 = {𝑎, 𝑏}} ∈ V)
15 abrexex2g 7945 . . . . 5 ((𝑉𝑊 ∧ ∀𝑏𝑉 {𝑝𝑝 = {𝑎, 𝑏}} ∈ V) → {𝑝 ∣ ∃𝑏𝑉 𝑝 = {𝑎, 𝑏}} ∈ V)
1614, 15mpdan 684 . . . 4 (𝑉𝑊 → {𝑝 ∣ ∃𝑏𝑉 𝑝 = {𝑎, 𝑏}} ∈ V)
1716ralrimivw 3142 . . 3 (𝑉𝑊 → ∀𝑎𝑉 {𝑝 ∣ ∃𝑏𝑉 𝑝 = {𝑎, 𝑏}} ∈ V)
18 abrexex2g 7945 . . 3 ((𝑉𝑊 ∧ ∀𝑎𝑉 {𝑝 ∣ ∃𝑏𝑉 𝑝 = {𝑎, 𝑏}} ∈ V) → {𝑝 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}} ∈ V)
1917, 18mpdan 684 . 2 (𝑉𝑊 → {𝑝 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}} ∈ V)
202, 7, 8, 19fvmptd 6996 1 (𝑉𝑊 → (Pairs‘𝑉) = {𝑝 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  ∃!weu 2554  {cab 2701  wral 3053  wrex 3062  Vcvv 3466  {cpr 4623  cmpt 5222  cfv 6534  Pairscspr 46655
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-iota 6486  df-fun 6536  df-fv 6542  df-spr 46656
This theorem is referenced by:  sprvalpw  46658  sprssspr  46659  prprspr2  46696
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