Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  prelspr Structured version   Visualization version   GIF version

Theorem prelspr 46452
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 5446 . . . . 5 ((𝑋𝑉𝑌𝑉) → {𝑋, 𝑌} ∈ 𝒫 𝑉)
2 eqidd 2731 . . . . . 6 ((𝑋𝑉𝑌𝑉) → {𝑋, 𝑌} = {𝑋, 𝑌})
3 preq1 4736 . . . . . . . 8 (𝑎 = 𝑋 → {𝑎, 𝑏} = {𝑋, 𝑏})
43eqeq2d 2741 . . . . . . 7 (𝑎 = 𝑋 → ({𝑋, 𝑌} = {𝑎, 𝑏} ↔ {𝑋, 𝑌} = {𝑋, 𝑏}))
5 preq2 4737 . . . . . . . 8 (𝑏 = 𝑌 → {𝑋, 𝑏} = {𝑋, 𝑌})
65eqeq2d 2741 . . . . . . 7 (𝑏 = 𝑌 → ({𝑋, 𝑌} = {𝑋, 𝑏} ↔ {𝑋, 𝑌} = {𝑋, 𝑌}))
74, 6rspc2ev 3623 . . . . . 6 ((𝑋𝑉𝑌𝑉 ∧ {𝑋, 𝑌} = {𝑋, 𝑌}) → ∃𝑎𝑉𝑏𝑉 {𝑋, 𝑌} = {𝑎, 𝑏})
82, 7mpd3an3 1460 . . . . 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 2734 . . . . 5 (𝑝 = {𝑋, 𝑌} → (𝑝 = {𝑎, 𝑏} ↔ {𝑋, 𝑌} = {𝑎, 𝑏}))
12112rexbidv 3217 . . . 4 (𝑝 = {𝑋, 𝑌} → (∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏} ↔ ∃𝑎𝑉𝑏𝑉 {𝑋, 𝑌} = {𝑎, 𝑏}))
1312elrab 3682 . . 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 46446 . . 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 2834 1 ((𝑉𝑊 ∧ (𝑋𝑉𝑌𝑉)) → {𝑋, 𝑌} ∈ (Pairs‘𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1539  wcel 2104  wrex 3068  {crab 3430  𝒫 cpw 4601  {cpr 4629  cfv 6542  Pairscspr 46443
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6494  df-fun 6544  df-fv 6550  df-spr 46444
This theorem is referenced by:  sprsymrelfolem2  46459  reupr  46488
  Copyright terms: Public domain W3C validator