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Theorem prelspr 47092
Description: An unordered pair of elements of a fixed set 𝑉 belongs to the set of all unordered pairs over the set 𝑉. (Contributed by AV, 21-Nov-2021.)
Assertion
Ref Expression
prelspr ((𝑉𝑊 ∧ (𝑋𝑉𝑌𝑉)) → {𝑋, 𝑌} ∈ (Pairs‘𝑉))

Proof of Theorem prelspr
Dummy variables 𝑎 𝑏 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prelpwi 5444 . . . . 5 ((𝑋𝑉𝑌𝑉) → {𝑋, 𝑌} ∈ 𝒫 𝑉)
2 eqidd 2727 . . . . . 6 ((𝑋𝑉𝑌𝑉) → {𝑋, 𝑌} = {𝑋, 𝑌})
3 preq1 4733 . . . . . . . 8 (𝑎 = 𝑋 → {𝑎, 𝑏} = {𝑋, 𝑏})
43eqeq2d 2737 . . . . . . 7 (𝑎 = 𝑋 → ({𝑋, 𝑌} = {𝑎, 𝑏} ↔ {𝑋, 𝑌} = {𝑋, 𝑏}))
5 preq2 4734 . . . . . . . 8 (𝑏 = 𝑌 → {𝑋, 𝑏} = {𝑋, 𝑌})
65eqeq2d 2737 . . . . . . 7 (𝑏 = 𝑌 → ({𝑋, 𝑌} = {𝑋, 𝑏} ↔ {𝑋, 𝑌} = {𝑋, 𝑌}))
74, 6rspc2ev 3621 . . . . . 6 ((𝑋𝑉𝑌𝑉 ∧ {𝑋, 𝑌} = {𝑋, 𝑌}) → ∃𝑎𝑉𝑏𝑉 {𝑋, 𝑌} = {𝑎, 𝑏})
82, 7mpd3an3 1459 . . . . 5 ((𝑋𝑉𝑌𝑉) → ∃𝑎𝑉𝑏𝑉 {𝑋, 𝑌} = {𝑎, 𝑏})
91, 8jca 510 . . . 4 ((𝑋𝑉𝑌𝑉) → ({𝑋, 𝑌} ∈ 𝒫 𝑉 ∧ ∃𝑎𝑉𝑏𝑉 {𝑋, 𝑌} = {𝑎, 𝑏}))
109adantl 480 . . 3 ((𝑉𝑊 ∧ (𝑋𝑉𝑌𝑉)) → ({𝑋, 𝑌} ∈ 𝒫 𝑉 ∧ ∃𝑎𝑉𝑏𝑉 {𝑋, 𝑌} = {𝑎, 𝑏}))
11 eqeq1 2730 . . . . 5 (𝑝 = {𝑋, 𝑌} → (𝑝 = {𝑎, 𝑏} ↔ {𝑋, 𝑌} = {𝑎, 𝑏}))
12112rexbidv 3210 . . . 4 (𝑝 = {𝑋, 𝑌} → (∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏} ↔ ∃𝑎𝑉𝑏𝑉 {𝑋, 𝑌} = {𝑎, 𝑏}))
1312elrab 3681 . . 3 ({𝑋, 𝑌} ∈ {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}} ↔ ({𝑋, 𝑌} ∈ 𝒫 𝑉 ∧ ∃𝑎𝑉𝑏𝑉 {𝑋, 𝑌} = {𝑎, 𝑏}))
1410, 13sylibr 233 . 2 ((𝑉𝑊 ∧ (𝑋𝑉𝑌𝑉)) → {𝑋, 𝑌} ∈ {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}})
15 sprvalpw 47086 . . 3 (𝑉𝑊 → (Pairs‘𝑉) = {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}})
1615adantr 479 . 2 ((𝑉𝑊 ∧ (𝑋𝑉𝑌𝑉)) → (Pairs‘𝑉) = {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}})
1714, 16eleqtrrd 2829 1 ((𝑉𝑊 ∧ (𝑋𝑉𝑌𝑉)) → {𝑋, 𝑌} ∈ (Pairs‘𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  wcel 2099  wrex 3060  {crab 3420  𝒫 cpw 4598  {cpr 4626  cfv 6544  Pairscspr 47083
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 2697  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pr 5424  ax-un 7736
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ral 3052  df-rex 3061  df-rab 3421  df-v 3465  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4907  df-iun 4996  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-iota 6496  df-fun 6546  df-fv 6552  df-spr 47084
This theorem is referenced by:  sprsymrelfolem2  47099  reupr  47128
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