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Theorem prelspr 41054
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 4886 . . . . 5 ((𝑋𝑉𝑌𝑉) → {𝑋, 𝑌} ∈ 𝒫 𝑉)
2 eqidd 2622 . . . . . 6 ((𝑋𝑉𝑌𝑉) → {𝑋, 𝑌} = {𝑋, 𝑌})
3 preq1 4245 . . . . . . . 8 (𝑎 = 𝑋 → {𝑎, 𝑏} = {𝑋, 𝑏})
43eqeq2d 2631 . . . . . . 7 (𝑎 = 𝑋 → ({𝑋, 𝑌} = {𝑎, 𝑏} ↔ {𝑋, 𝑌} = {𝑋, 𝑏}))
5 preq2 4246 . . . . . . . 8 (𝑏 = 𝑌 → {𝑋, 𝑏} = {𝑋, 𝑌})
65eqeq2d 2631 . . . . . . 7 (𝑏 = 𝑌 → ({𝑋, 𝑌} = {𝑋, 𝑏} ↔ {𝑋, 𝑌} = {𝑋, 𝑌}))
74, 6rspc2ev 3313 . . . . . 6 ((𝑋𝑉𝑌𝑉 ∧ {𝑋, 𝑌} = {𝑋, 𝑌}) → ∃𝑎𝑉𝑏𝑉 {𝑋, 𝑌} = {𝑎, 𝑏})
82, 7mpd3an3 1422 . . . . 5 ((𝑋𝑉𝑌𝑉) → ∃𝑎𝑉𝑏𝑉 {𝑋, 𝑌} = {𝑎, 𝑏})
91, 8jca 554 . . . 4 ((𝑋𝑉𝑌𝑉) → ({𝑋, 𝑌} ∈ 𝒫 𝑉 ∧ ∃𝑎𝑉𝑏𝑉 {𝑋, 𝑌} = {𝑎, 𝑏}))
109adantl 482 . . 3 ((𝑉𝑊 ∧ (𝑋𝑉𝑌𝑉)) → ({𝑋, 𝑌} ∈ 𝒫 𝑉 ∧ ∃𝑎𝑉𝑏𝑉 {𝑋, 𝑌} = {𝑎, 𝑏}))
11 eqeq1 2625 . . . . 5 (𝑝 = {𝑋, 𝑌} → (𝑝 = {𝑎, 𝑏} ↔ {𝑋, 𝑌} = {𝑎, 𝑏}))
12112rexbidv 3052 . . . 4 (𝑝 = {𝑋, 𝑌} → (∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏} ↔ ∃𝑎𝑉𝑏𝑉 {𝑋, 𝑌} = {𝑎, 𝑏}))
1312elrab 3351 . . 3 ({𝑋, 𝑌} ∈ {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}} ↔ ({𝑋, 𝑌} ∈ 𝒫 𝑉 ∧ ∃𝑎𝑉𝑏𝑉 {𝑋, 𝑌} = {𝑎, 𝑏}))
1410, 13sylibr 224 . 2 ((𝑉𝑊 ∧ (𝑋𝑉𝑌𝑉)) → {𝑋, 𝑌} ∈ {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}})
15 sprvalpw 41048 . . 3 (𝑉𝑊 → (Pairs‘𝑉) = {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}})
1615adantr 481 . 2 ((𝑉𝑊 ∧ (𝑋𝑉𝑌𝑉)) → (Pairs‘𝑉) = {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}})
1714, 16eleqtrrd 2701 1 ((𝑉𝑊 ∧ (𝑋𝑉𝑌𝑉)) → {𝑋, 𝑌} ∈ (Pairs‘𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wrex 2909  {crab 2912  𝒫 cpw 4136  {cpr 4157  cfv 5857  Pairscspr 41045
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-spr 41046
This theorem is referenced by:  sprsymrelfolem2  41061
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