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 42246
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 5064 . . . . 5 ((𝑋𝑉𝑌𝑉) → {𝑋, 𝑌} ∈ 𝒫 𝑉)
2 eqidd 2761 . . . . . 6 ((𝑋𝑉𝑌𝑉) → {𝑋, 𝑌} = {𝑋, 𝑌})
3 preq1 4412 . . . . . . . 8 (𝑎 = 𝑋 → {𝑎, 𝑏} = {𝑋, 𝑏})
43eqeq2d 2770 . . . . . . 7 (𝑎 = 𝑋 → ({𝑋, 𝑌} = {𝑎, 𝑏} ↔ {𝑋, 𝑌} = {𝑋, 𝑏}))
5 preq2 4413 . . . . . . . 8 (𝑏 = 𝑌 → {𝑋, 𝑏} = {𝑋, 𝑌})
65eqeq2d 2770 . . . . . . 7 (𝑏 = 𝑌 → ({𝑋, 𝑌} = {𝑋, 𝑏} ↔ {𝑋, 𝑌} = {𝑋, 𝑌}))
74, 6rspc2ev 3463 . . . . . 6 ((𝑋𝑉𝑌𝑉 ∧ {𝑋, 𝑌} = {𝑋, 𝑌}) → ∃𝑎𝑉𝑏𝑉 {𝑋, 𝑌} = {𝑎, 𝑏})
82, 7mpd3an3 1574 . . . . 5 ((𝑋𝑉𝑌𝑉) → ∃𝑎𝑉𝑏𝑉 {𝑋, 𝑌} = {𝑎, 𝑏})
91, 8jca 555 . . . 4 ((𝑋𝑉𝑌𝑉) → ({𝑋, 𝑌} ∈ 𝒫 𝑉 ∧ ∃𝑎𝑉𝑏𝑉 {𝑋, 𝑌} = {𝑎, 𝑏}))
109adantl 473 . . 3 ((𝑉𝑊 ∧ (𝑋𝑉𝑌𝑉)) → ({𝑋, 𝑌} ∈ 𝒫 𝑉 ∧ ∃𝑎𝑉𝑏𝑉 {𝑋, 𝑌} = {𝑎, 𝑏}))
11 eqeq1 2764 . . . . 5 (𝑝 = {𝑋, 𝑌} → (𝑝 = {𝑎, 𝑏} ↔ {𝑋, 𝑌} = {𝑎, 𝑏}))
12112rexbidv 3195 . . . 4 (𝑝 = {𝑋, 𝑌} → (∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏} ↔ ∃𝑎𝑉𝑏𝑉 {𝑋, 𝑌} = {𝑎, 𝑏}))
1312elrab 3504 . . 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 42240 . . 3 (𝑉𝑊 → (Pairs‘𝑉) = {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}})
1615adantr 472 . 2 ((𝑉𝑊 ∧ (𝑋𝑉𝑌𝑉)) → (Pairs‘𝑉) = {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}})
1714, 16eleqtrrd 2842 1 ((𝑉𝑊 ∧ (𝑋𝑉𝑌𝑉)) → {𝑋, 𝑌} ∈ (Pairs‘𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  wrex 3051  {crab 3054  𝒫 cpw 4302  {cpr 4323  cfv 6049  Pairscspr 42237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-spr 42238
This theorem is referenced by:  sprsymrelfolem2  42253
  Copyright terms: Public domain W3C validator