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Theorem sprval 43862
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 43861 . . 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 3398 . . . . 5 (𝑣 = 𝑉 → (∃𝑏𝑣 𝑝 = {𝑎, 𝑏} ↔ ∃𝑏𝑉 𝑝 = {𝑎, 𝑏}))
53, 4rexeqbidv 3394 . . . 4 (𝑣 = 𝑉 → (∃𝑎𝑣𝑏𝑣 𝑝 = {𝑎, 𝑏} ↔ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}))
65adantl 485 . . 3 ((𝑉𝑊𝑣 = 𝑉) → (∃𝑎𝑣𝑏𝑣 𝑝 = {𝑎, 𝑏} ↔ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}))
76abbidv 2888 . 2 ((𝑉𝑊𝑣 = 𝑉) → {𝑝 ∣ ∃𝑎𝑣𝑏𝑣 𝑝 = {𝑎, 𝑏}} = {𝑝 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}})
8 elex 3498 . 2 (𝑉𝑊𝑉 ∈ V)
9 zfpair2 5319 . . . . . . . 8 {𝑎, 𝑏} ∈ V
10 eueq 3685 . . . . . . . 8 ({𝑎, 𝑏} ∈ V ↔ ∃!𝑝 𝑝 = {𝑎, 𝑏})
119, 10mpbi 233 . . . . . . 7 ∃!𝑝 𝑝 = {𝑎, 𝑏}
12 euabex 5341 . . . . . . 7 (∃!𝑝 𝑝 = {𝑎, 𝑏} → {𝑝𝑝 = {𝑎, 𝑏}} ∈ V)
1311, 12mp1i 13 . . . . . 6 (𝑉𝑊 → {𝑝𝑝 = {𝑎, 𝑏}} ∈ V)
1413ralrimivw 3178 . . . . 5 (𝑉𝑊 → ∀𝑏𝑉 {𝑝𝑝 = {𝑎, 𝑏}} ∈ V)
15 abrexex2g 7657 . . . . 5 ((𝑉𝑊 ∧ ∀𝑏𝑉 {𝑝𝑝 = {𝑎, 𝑏}} ∈ V) → {𝑝 ∣ ∃𝑏𝑉 𝑝 = {𝑎, 𝑏}} ∈ V)
1614, 15mpdan 686 . . . 4 (𝑉𝑊 → {𝑝 ∣ ∃𝑏𝑉 𝑝 = {𝑎, 𝑏}} ∈ V)
1716ralrimivw 3178 . . 3 (𝑉𝑊 → ∀𝑎𝑉 {𝑝 ∣ ∃𝑏𝑉 𝑝 = {𝑎, 𝑏}} ∈ V)
18 abrexex2g 7657 . . 3 ((𝑉𝑊 ∧ ∀𝑎𝑉 {𝑝 ∣ ∃𝑏𝑉 𝑝 = {𝑎, 𝑏}} ∈ V) → {𝑝 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}} ∈ V)
1917, 18mpdan 686 . 2 (𝑉𝑊 → {𝑝 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}} ∈ V)
202, 7, 8, 19fvmptd 6764 1 (𝑉𝑊 → (Pairs‘𝑉) = {𝑝 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  ∃!weu 2654  {cab 2802  wral 3133  wrex 3134  Vcvv 3480  {cpr 4552  cmpt 5133  cfv 6344  Pairscspr 43860
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pr 5318  ax-un 7452
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-spr 43861
This theorem is referenced by:  sprvalpw  43863  sprssspr  43864  prprspr2  43901
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