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Theorem prprval 43957
Description: The set of all proper unordered pairs over a given set 𝑉. (Contributed by AV, 29-Apr-2023.)
Assertion
Ref Expression
prprval (𝑉𝑊 → (Pairsproper𝑉) = {𝑝 ∣ ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏})})
Distinct variable groups:   𝑉,𝑎,𝑏,𝑝   𝑊,𝑎,𝑏,𝑝

Proof of Theorem prprval
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 df-prpr 43956 . 2 Pairsproper = (𝑣 ∈ V ↦ {𝑝 ∣ ∃𝑎𝑣𝑏𝑣 (𝑎𝑏𝑝 = {𝑎, 𝑏})})
2 rexeq 3397 . . . . 5 (𝑣 = 𝑉 → (∃𝑏𝑣 (𝑎𝑏𝑝 = {𝑎, 𝑏}) ↔ ∃𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏})))
32rexeqbi1dv 3395 . . . 4 (𝑣 = 𝑉 → (∃𝑎𝑣𝑏𝑣 (𝑎𝑏𝑝 = {𝑎, 𝑏}) ↔ ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏})))
43abbidv 2888 . . 3 (𝑣 = 𝑉 → {𝑝 ∣ ∃𝑎𝑣𝑏𝑣 (𝑎𝑏𝑝 = {𝑎, 𝑏})} = {𝑝 ∣ ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏})})
54adantl 485 . 2 ((𝑉𝑊𝑣 = 𝑉) → {𝑝 ∣ ∃𝑎𝑣𝑏𝑣 (𝑎𝑏𝑝 = {𝑎, 𝑏})} = {𝑝 ∣ ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏})})
6 elex 3498 . 2 (𝑉𝑊𝑉 ∈ V)
7 simpr 488 . . . . . . . 8 ((𝑎𝑏𝑝 = {𝑎, 𝑏}) → 𝑝 = {𝑎, 𝑏})
87ss2abi 4029 . . . . . . 7 {𝑝 ∣ (𝑎𝑏𝑝 = {𝑎, 𝑏})} ⊆ {𝑝𝑝 = {𝑎, 𝑏}}
9 zfpair2 5318 . . . . . . . . 9 {𝑎, 𝑏} ∈ V
109eueqi 3686 . . . . . . . 8 ∃!𝑝 𝑝 = {𝑎, 𝑏}
11 euabex 5340 . . . . . . . 8 (∃!𝑝 𝑝 = {𝑎, 𝑏} → {𝑝𝑝 = {𝑎, 𝑏}} ∈ V)
1210, 11mp1i 13 . . . . . . 7 (𝑉𝑊 → {𝑝𝑝 = {𝑎, 𝑏}} ∈ V)
13 ssexg 5213 . . . . . . 7 (({𝑝 ∣ (𝑎𝑏𝑝 = {𝑎, 𝑏})} ⊆ {𝑝𝑝 = {𝑎, 𝑏}} ∧ {𝑝𝑝 = {𝑎, 𝑏}} ∈ V) → {𝑝 ∣ (𝑎𝑏𝑝 = {𝑎, 𝑏})} ∈ V)
148, 12, 13sylancr 590 . . . . . 6 (𝑉𝑊 → {𝑝 ∣ (𝑎𝑏𝑝 = {𝑎, 𝑏})} ∈ V)
1514ralrimivw 3178 . . . . 5 (𝑉𝑊 → ∀𝑏𝑉 {𝑝 ∣ (𝑎𝑏𝑝 = {𝑎, 𝑏})} ∈ V)
16 abrexex2g 7660 . . . . 5 ((𝑉𝑊 ∧ ∀𝑏𝑉 {𝑝 ∣ (𝑎𝑏𝑝 = {𝑎, 𝑏})} ∈ V) → {𝑝 ∣ ∃𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏})} ∈ V)
1715, 16mpdan 686 . . . 4 (𝑉𝑊 → {𝑝 ∣ ∃𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏})} ∈ V)
1817ralrimivw 3178 . . 3 (𝑉𝑊 → ∀𝑎𝑉 {𝑝 ∣ ∃𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏})} ∈ V)
19 abrexex2g 7660 . . 3 ((𝑉𝑊 ∧ ∀𝑎𝑉 {𝑝 ∣ ∃𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏})} ∈ V) → {𝑝 ∣ ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏})} ∈ V)
2018, 19mpdan 686 . 2 (𝑉𝑊 → {𝑝 ∣ ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏})} ∈ V)
211, 5, 6, 20fvmptd2 6767 1 (𝑉𝑊 → (Pairsproper𝑉) = {𝑝 ∣ ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏})})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  ∃!weu 2654  {cab 2802  wne 3014  wral 3133  wrex 3134  Vcvv 3480  wss 3919  {cpr 4552  cfv 6343  Pairspropercprpr 43955
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-prpr 43956
This theorem is referenced by:  prprvalpw  43958  prprspr2  43961
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