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Theorem prprval 47023
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 47022 . 2 Pairsproper = (𝑣 ∈ V ↦ {𝑝 ∣ ∃𝑎𝑣𝑏𝑣 (𝑎𝑏𝑝 = {𝑎, 𝑏})})
2 rexeq 3310 . . . . 5 (𝑣 = 𝑉 → (∃𝑏𝑣 (𝑎𝑏𝑝 = {𝑎, 𝑏}) ↔ ∃𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏})))
32rexeqbi1dv 3323 . . . 4 (𝑣 = 𝑉 → (∃𝑎𝑣𝑏𝑣 (𝑎𝑏𝑝 = {𝑎, 𝑏}) ↔ ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏})))
43abbidv 2794 . . 3 (𝑣 = 𝑉 → {𝑝 ∣ ∃𝑎𝑣𝑏𝑣 (𝑎𝑏𝑝 = {𝑎, 𝑏})} = {𝑝 ∣ ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏})})
54adantl 480 . 2 ((𝑉𝑊𝑣 = 𝑉) → {𝑝 ∣ ∃𝑎𝑣𝑏𝑣 (𝑎𝑏𝑝 = {𝑎, 𝑏})} = {𝑝 ∣ ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏})})
6 elex 3481 . 2 (𝑉𝑊𝑉 ∈ V)
7 simpr 483 . . . . . . . 8 ((𝑎𝑏𝑝 = {𝑎, 𝑏}) → 𝑝 = {𝑎, 𝑏})
87ss2abi 4061 . . . . . . 7 {𝑝 ∣ (𝑎𝑏𝑝 = {𝑎, 𝑏})} ⊆ {𝑝𝑝 = {𝑎, 𝑏}}
9 zfpair2 5433 . . . . . . . . 9 {𝑎, 𝑏} ∈ V
109eueqi 3702 . . . . . . . 8 ∃!𝑝 𝑝 = {𝑎, 𝑏}
11 euabex 5466 . . . . . . . 8 (∃!𝑝 𝑝 = {𝑎, 𝑏} → {𝑝𝑝 = {𝑎, 𝑏}} ∈ V)
1210, 11mp1i 13 . . . . . . 7 (𝑉𝑊 → {𝑝𝑝 = {𝑎, 𝑏}} ∈ V)
13 ssexg 5327 . . . . . . 7 (({𝑝 ∣ (𝑎𝑏𝑝 = {𝑎, 𝑏})} ⊆ {𝑝𝑝 = {𝑎, 𝑏}} ∧ {𝑝𝑝 = {𝑎, 𝑏}} ∈ V) → {𝑝 ∣ (𝑎𝑏𝑝 = {𝑎, 𝑏})} ∈ V)
148, 12, 13sylancr 585 . . . . . 6 (𝑉𝑊 → {𝑝 ∣ (𝑎𝑏𝑝 = {𝑎, 𝑏})} ∈ V)
1514ralrimivw 3139 . . . . 5 (𝑉𝑊 → ∀𝑏𝑉 {𝑝 ∣ (𝑎𝑏𝑝 = {𝑎, 𝑏})} ∈ V)
16 abrexex2g 7977 . . . . 5 ((𝑉𝑊 ∧ ∀𝑏𝑉 {𝑝 ∣ (𝑎𝑏𝑝 = {𝑎, 𝑏})} ∈ V) → {𝑝 ∣ ∃𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏})} ∈ V)
1715, 16mpdan 685 . . . 4 (𝑉𝑊 → {𝑝 ∣ ∃𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏})} ∈ V)
1817ralrimivw 3139 . . 3 (𝑉𝑊 → ∀𝑎𝑉 {𝑝 ∣ ∃𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏})} ∈ V)
19 abrexex2g 7977 . . 3 ((𝑉𝑊 ∧ ∀𝑎𝑉 {𝑝 ∣ ∃𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏})} ∈ V) → {𝑝 ∣ ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏})} ∈ V)
2018, 19mpdan 685 . 2 (𝑉𝑊 → {𝑝 ∣ ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏})} ∈ V)
211, 5, 6, 20fvmptd2 7016 1 (𝑉𝑊 → (Pairsproper𝑉) = {𝑝 ∣ ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑝 = {𝑎, 𝑏})})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  ∃!weu 2556  {cab 2702  wne 2929  wral 3050  wrex 3059  Vcvv 3461  wss 3946  {cpr 4634  cfv 6553  Pairspropercprpr 47021
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pr 5432  ax-un 7745
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4325  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5579  df-xp 5687  df-rel 5688  df-cnv 5689  df-co 5690  df-dm 5691  df-iota 6505  df-fun 6555  df-fv 6561  df-prpr 47022
This theorem is referenced by:  prprvalpw  47024  prprspr2  47027
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