Theorem prprval 48118

Description: The set of all proper unordered pairs over a given set 𝑉. (Contributed by AV, 29-Apr-2023.)

Assertion

Ref	Expression
prprval	⊢ (𝑉 ∈ 𝑊 → (Pairsproper‘𝑉) = {𝑝 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑝 = {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.

Step	Hyp	Ref	Expression
1		df-prpr 48117	. 2 ⊢ Pairsproper = (𝑣 ∈ V ↦ {𝑝 ∣ ∃𝑎 ∈ 𝑣 ∃𝑏 ∈ 𝑣 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})})
2		rexeq 3319	. . . . 5 ⊢ (𝑣 = 𝑉 → (∃𝑏 ∈ 𝑣 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏}) ↔ ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})))
3	2	rexeqbi1dv 3334	. . . 4 ⊢ (𝑣 = 𝑉 → (∃𝑎 ∈ 𝑣 ∃𝑏 ∈ 𝑣 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏}) ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})))
4	3	abbidv 2831	. . 3 ⊢ (𝑣 = 𝑉 → {𝑝 ∣ ∃𝑎 ∈ 𝑣 ∃𝑏 ∈ 𝑣 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})} = {𝑝 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})})
5	4	adantl 486	. 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑣 = 𝑉) → {𝑝 ∣ ∃𝑎 ∈ 𝑣 ∃𝑏 ∈ 𝑣 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})} = {𝑝 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})})
6		elex 3478	. 2 ⊢ (𝑉 ∈ 𝑊 → 𝑉 ∈ V)
7		simpr 489	. . . . . . . 8 ⊢ ((𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏}) → 𝑝 = {𝑎, 𝑏})
8	7	ss2abi 4022	. . . . . . 7 ⊢ {𝑝 ∣ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})} ⊆ {𝑝 ∣ 𝑝 = {𝑎, 𝑏}}
9		zfpair2 5396	. . . . . . . . 9 ⊢ {𝑎, 𝑏} ∈ V
10	9	eueqi 3675	. . . . . . . 8 ⊢ ∃!𝑝 𝑝 = {𝑎, 𝑏}
11		euabex 5433	. . . . . . . 8 ⊢ (∃!𝑝 𝑝 = {𝑎, 𝑏} → {𝑝 ∣ 𝑝 = {𝑎, 𝑏}} ∈ V)
12	10, 11	mp1i 14	. . . . . . 7 ⊢ (𝑉 ∈ 𝑊 → {𝑝 ∣ 𝑝 = {𝑎, 𝑏}} ∈ V)
13		ssexg 5284	. . . . . . 7 ⊢ (({𝑝 ∣ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})} ⊆ {𝑝 ∣ 𝑝 = {𝑎, 𝑏}} ∧ {𝑝 ∣ 𝑝 = {𝑎, 𝑏}} ∈ V) → {𝑝 ∣ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})} ∈ V)
14	8, 12, 13	sylancr 598	. . . . . 6 ⊢ (𝑉 ∈ 𝑊 → {𝑝 ∣ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})} ∈ V)
15	14	ralrimivw 3161	. . . . 5 ⊢ (𝑉 ∈ 𝑊 → ∀𝑏 ∈ 𝑉 {𝑝 ∣ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})} ∈ V)
16		abrexex2g 7949	. . . . 5 ⊢ ((𝑉 ∈ 𝑊 ∧ ∀𝑏 ∈ 𝑉 {𝑝 ∣ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})} ∈ V) → {𝑝 ∣ ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})} ∈ V)
17	15, 16	mpdan 699	. . . 4 ⊢ (𝑉 ∈ 𝑊 → {𝑝 ∣ ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})} ∈ V)
18	17	ralrimivw 3161	. . 3 ⊢ (𝑉 ∈ 𝑊 → ∀𝑎 ∈ 𝑉 {𝑝 ∣ ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})} ∈ V)
19		abrexex2g 7949	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ ∀𝑎 ∈ 𝑉 {𝑝 ∣ ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})} ∈ V) → {𝑝 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})} ∈ V)
20	18, 19	mpdan 699	. 2 ⊢ (𝑉 ∈ 𝑊 → {𝑝 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})} ∈ V)
21	1, 5, 6, 20	fvmptd2 6988	1 ⊢ (𝑉 ∈ 𝑊 → (Pairsproper‘𝑉) = {𝑝 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})})

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145  ∃!weu 2598  {cab 2743   ≠ wne 2960  ∀wral 3079  ∃wrex 3089  Vcvv 3457   ⊆ wss 3907  {cpr 4587  ‘cfv 6525  Pairs_propercprpr 48116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-iota 6481  df-fun 6527  df-fv 6533  df-prpr 48117

This theorem is referenced by:  prprvalpw  48119  prprspr2  48122