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Theorem sprval 46810
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 46809 . . 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 3317 . . . . 5 (𝑣 = 𝑉 → (∃𝑏𝑣 𝑝 = {𝑎, 𝑏} ↔ ∃𝑏𝑉 𝑝 = {𝑎, 𝑏}))
53, 4rexeqbidv 3339 . . . 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 2797 . 2 ((𝑉𝑊𝑣 = 𝑉) → {𝑝 ∣ ∃𝑎𝑣𝑏𝑣 𝑝 = {𝑎, 𝑏}} = {𝑝 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}})
8 elex 3489 . 2 (𝑉𝑊𝑉 ∈ V)
9 zfpair2 5425 . . . . . . . 8 {𝑎, 𝑏} ∈ V
10 eueq 3702 . . . . . . . 8 ({𝑎, 𝑏} ∈ V ↔ ∃!𝑝 𝑝 = {𝑎, 𝑏})
119, 10mpbi 229 . . . . . . 7 ∃!𝑝 𝑝 = {𝑎, 𝑏}
12 euabex 5458 . . . . . . 7 (∃!𝑝 𝑝 = {𝑎, 𝑏} → {𝑝𝑝 = {𝑎, 𝑏}} ∈ V)
1311, 12mp1i 13 . . . . . 6 (𝑉𝑊 → {𝑝𝑝 = {𝑎, 𝑏}} ∈ V)
1413ralrimivw 3146 . . . . 5 (𝑉𝑊 → ∀𝑏𝑉 {𝑝𝑝 = {𝑎, 𝑏}} ∈ V)
15 abrexex2g 7963 . . . . 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 3146 . . 3 (𝑉𝑊 → ∀𝑎𝑉 {𝑝 ∣ ∃𝑏𝑉 𝑝 = {𝑎, 𝑏}} ∈ V)
18 abrexex2g 7963 . . 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 7007 1 (𝑉𝑊 → (Pairs‘𝑉) = {𝑝 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1534  wcel 2099  ∃!weu 2558  {cab 2705  wral 3057  wrex 3066  Vcvv 3470  {cpr 4627  cmpt 5226  cfv 6543  Pairscspr 46808
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pr 5424  ax-un 7735
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-iun 4994  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-iota 6495  df-fun 6545  df-fv 6551  df-spr 46809
This theorem is referenced by:  sprvalpw  46811  sprssspr  46812  prprspr2  46849
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