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Theorem sprval 47404
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 47403 . . 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 3320 . . . . 5 (𝑣 = 𝑉 → (∃𝑏𝑣 𝑝 = {𝑎, 𝑏} ↔ ∃𝑏𝑉 𝑝 = {𝑎, 𝑏}))
53, 4rexeqbidv 3345 . . . 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 2806 . 2 ((𝑉𝑊𝑣 = 𝑉) → {𝑝 ∣ ∃𝑎𝑣𝑏𝑣 𝑝 = {𝑎, 𝑏}} = {𝑝 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}})
8 elex 3499 . 2 (𝑉𝑊𝑉 ∈ V)
9 zfpair2 5439 . . . . . . . 8 {𝑎, 𝑏} ∈ V
10 eueq 3717 . . . . . . . 8 ({𝑎, 𝑏} ∈ V ↔ ∃!𝑝 𝑝 = {𝑎, 𝑏})
119, 10mpbi 230 . . . . . . 7 ∃!𝑝 𝑝 = {𝑎, 𝑏}
12 euabex 5472 . . . . . . 7 (∃!𝑝 𝑝 = {𝑎, 𝑏} → {𝑝𝑝 = {𝑎, 𝑏}} ∈ V)
1311, 12mp1i 13 . . . . . 6 (𝑉𝑊 → {𝑝𝑝 = {𝑎, 𝑏}} ∈ V)
1413ralrimivw 3148 . . . . 5 (𝑉𝑊 → ∀𝑏𝑉 {𝑝𝑝 = {𝑎, 𝑏}} ∈ V)
15 abrexex2g 7988 . . . . 5 ((𝑉𝑊 ∧ ∀𝑏𝑉 {𝑝𝑝 = {𝑎, 𝑏}} ∈ V) → {𝑝 ∣ ∃𝑏𝑉 𝑝 = {𝑎, 𝑏}} ∈ V)
1614, 15mpdan 687 . . . 4 (𝑉𝑊 → {𝑝 ∣ ∃𝑏𝑉 𝑝 = {𝑎, 𝑏}} ∈ V)
1716ralrimivw 3148 . . 3 (𝑉𝑊 → ∀𝑎𝑉 {𝑝 ∣ ∃𝑏𝑉 𝑝 = {𝑎, 𝑏}} ∈ V)
18 abrexex2g 7988 . . 3 ((𝑉𝑊 ∧ ∀𝑎𝑉 {𝑝 ∣ ∃𝑏𝑉 𝑝 = {𝑎, 𝑏}} ∈ V) → {𝑝 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}} ∈ V)
1917, 18mpdan 687 . 2 (𝑉𝑊 → {𝑝 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}} ∈ V)
202, 7, 8, 19fvmptd 7023 1 (𝑉𝑊 → (Pairs‘𝑉) = {𝑝 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  ∃!weu 2566  {cab 2712  wral 3059  wrex 3068  Vcvv 3478  {cpr 4633  cmpt 5231  cfv 6563  Pairscspr 47402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-spr 47403
This theorem is referenced by:  sprvalpw  47405  sprssspr  47406  prprspr2  47443
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