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Theorem prprval 43078
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 43077 . 2 Pairsproper = (𝑣 ∈ V ↦ {𝑝 ∣ ∃𝑎𝑣𝑏𝑣 (𝑎𝑏𝑝 = {𝑎, 𝑏})})
2 rexeq 3339 . . . . 5 (𝑣 = 𝑉 → (∃𝑏𝑣 (𝑎𝑏𝑝 = {𝑎, 𝑏}) ↔ ∃𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏})))
32rexeqbi1dv 3337 . . . 4 (𝑣 = 𝑉 → (∃𝑎𝑣𝑏𝑣 (𝑎𝑏𝑝 = {𝑎, 𝑏}) ↔ ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏})))
43abbidv 2836 . . 3 (𝑣 = 𝑉 → {𝑝 ∣ ∃𝑎𝑣𝑏𝑣 (𝑎𝑏𝑝 = {𝑎, 𝑏})} = {𝑝 ∣ ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏})})
54adantl 474 . 2 ((𝑉𝑊𝑣 = 𝑉) → {𝑝 ∣ ∃𝑎𝑣𝑏𝑣 (𝑎𝑏𝑝 = {𝑎, 𝑏})} = {𝑝 ∣ ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏})})
6 elex 3426 . 2 (𝑉𝑊𝑉 ∈ V)
7 simpr 477 . . . . . . . 8 ((𝑎𝑏𝑝 = {𝑎, 𝑏}) → 𝑝 = {𝑎, 𝑏})
87ss2abi 3926 . . . . . . 7 {𝑝 ∣ (𝑎𝑏𝑝 = {𝑎, 𝑏})} ⊆ {𝑝𝑝 = {𝑎, 𝑏}}
9 zfpair2 5183 . . . . . . . . 9 {𝑎, 𝑏} ∈ V
109eueqi 3607 . . . . . . . 8 ∃!𝑝 𝑝 = {𝑎, 𝑏}
11 euabex 5206 . . . . . . . 8 (∃!𝑝 𝑝 = {𝑎, 𝑏} → {𝑝𝑝 = {𝑎, 𝑏}} ∈ V)
1210, 11mp1i 13 . . . . . . 7 (𝑉𝑊 → {𝑝𝑝 = {𝑎, 𝑏}} ∈ V)
13 ssexg 5079 . . . . . . 7 (({𝑝 ∣ (𝑎𝑏𝑝 = {𝑎, 𝑏})} ⊆ {𝑝𝑝 = {𝑎, 𝑏}} ∧ {𝑝𝑝 = {𝑎, 𝑏}} ∈ V) → {𝑝 ∣ (𝑎𝑏𝑝 = {𝑎, 𝑏})} ∈ V)
148, 12, 13sylancr 579 . . . . . 6 (𝑉𝑊 → {𝑝 ∣ (𝑎𝑏𝑝 = {𝑎, 𝑏})} ∈ V)
1514ralrimivw 3126 . . . . 5 (𝑉𝑊 → ∀𝑏𝑉 {𝑝 ∣ (𝑎𝑏𝑝 = {𝑎, 𝑏})} ∈ V)
16 abrexex2g 7475 . . . . 5 ((𝑉𝑊 ∧ ∀𝑏𝑉 {𝑝 ∣ (𝑎𝑏𝑝 = {𝑎, 𝑏})} ∈ V) → {𝑝 ∣ ∃𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏})} ∈ V)
1715, 16mpdan 675 . . . 4 (𝑉𝑊 → {𝑝 ∣ ∃𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏})} ∈ V)
1817ralrimivw 3126 . . 3 (𝑉𝑊 → ∀𝑎𝑉 {𝑝 ∣ ∃𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏})} ∈ V)
19 abrexex2g 7475 . . 3 ((𝑉𝑊 ∧ ∀𝑎𝑉 {𝑝 ∣ ∃𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏})} ∈ V) → {𝑝 ∣ ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏})} ∈ V)
2018, 19mpdan 675 . 2 (𝑉𝑊 → {𝑝 ∣ ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏})} ∈ V)
211, 5, 6, 20fvmptd2 6600 1 (𝑉𝑊 → (Pairsproper𝑉) = {𝑝 ∣ ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏})})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387   = wceq 1508  wcel 2051  ∃!weu 2584  {cab 2751  wne 2960  wral 3081  wrex 3082  Vcvv 3408  wss 3822  {cpr 4437  cfv 6185  Pairspropercprpr 43076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-rep 5045  ax-sep 5056  ax-nul 5063  ax-pr 5182  ax-un 7277
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-ral 3086  df-rex 3087  df-reu 3088  df-rab 3090  df-v 3410  df-sbc 3675  df-csb 3780  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-iun 4790  df-br 4926  df-opab 4988  df-mpt 5005  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-prpr 43077
This theorem is referenced by:  prprvalpw  43079  prprspr2  43082
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