Theorem sprval 48256

Description: The set of all unordered pairs over a given set 𝑉. (Contributed by AV, 21-Nov-2021.)

Assertion

Ref	Expression
sprval	⊢ (𝑉 ∈ 𝑊 → (Pairs‘𝑉) = {𝑝 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}})

Distinct variable groups:   𝑉,𝑎,𝑏,𝑝   𝑊,𝑎,𝑏,𝑝

Proof of Theorem sprval

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-spr 48255	. . 3 ⊢ Pairs = (𝑣 ∈ V ↦ {𝑝 ∣ ∃𝑎 ∈ 𝑣 ∃𝑏 ∈ 𝑣 𝑝 = {𝑎, 𝑏}})
2	1	a1i 11	. 2 ⊢ (𝑉 ∈ 𝑊 → Pairs = (𝑣 ∈ V ↦ {𝑝 ∣ ∃𝑎 ∈ 𝑣 ∃𝑏 ∈ 𝑣 𝑝 = {𝑎, 𝑏}}))
3		id 23	. . . . 5 ⊢ (𝑣 = 𝑉 → 𝑣 = 𝑉)
4		rexeq 3318	. . . . 5 ⊢ (𝑣 = 𝑉 → (∃𝑏 ∈ 𝑣 𝑝 = {𝑎, 𝑏} ↔ ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}))
5	3, 4	rexeqbidv 3338	. . . 4 ⊢ (𝑣 = 𝑉 → (∃𝑎 ∈ 𝑣 ∃𝑏 ∈ 𝑣 𝑝 = {𝑎, 𝑏} ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}))
6	5	adantl 486	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑣 = 𝑉) → (∃𝑎 ∈ 𝑣 ∃𝑏 ∈ 𝑣 𝑝 = {𝑎, 𝑏} ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}))
7	6	abbidv 2828	. 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑣 = 𝑉) → {𝑝 ∣ ∃𝑎 ∈ 𝑣 ∃𝑏 ∈ 𝑣 𝑝 = {𝑎, 𝑏}} = {𝑝 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}})
8		elex 3475	. 2 ⊢ (𝑉 ∈ 𝑊 → 𝑉 ∈ V)
9		zfpair2 5404	. . . . . . . 8 ⊢ {𝑎, 𝑏} ∈ V
10		eueq 3670	. . . . . . . 8 ⊢ ({𝑎, 𝑏} ∈ V ↔ ∃!𝑝 𝑝 = {𝑎, 𝑏})
11	9, 10	mpbi 233	. . . . . . 7 ⊢ ∃!𝑝 𝑝 = {𝑎, 𝑏}
12		euabex 5441	. . . . . . 7 ⊢ (∃!𝑝 𝑝 = {𝑎, 𝑏} → {𝑝 ∣ 𝑝 = {𝑎, 𝑏}} ∈ V)
13	11, 12	mp1i 14	. . . . . 6 ⊢ (𝑉 ∈ 𝑊 → {𝑝 ∣ 𝑝 = {𝑎, 𝑏}} ∈ V)
14	13	ralrimivw 3160	. . . . 5 ⊢ (𝑉 ∈ 𝑊 → ∀𝑏 ∈ 𝑉 {𝑝 ∣ 𝑝 = {𝑎, 𝑏}} ∈ V)
15		abrexex2g 7959	. . . . 5 ⊢ ((𝑉 ∈ 𝑊 ∧ ∀𝑏 ∈ 𝑉 {𝑝 ∣ 𝑝 = {𝑎, 𝑏}} ∈ V) → {𝑝 ∣ ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}} ∈ V)
16	14, 15	mpdan 699	. . . 4 ⊢ (𝑉 ∈ 𝑊 → {𝑝 ∣ ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}} ∈ V)
17	16	ralrimivw 3160	. . 3 ⊢ (𝑉 ∈ 𝑊 → ∀𝑎 ∈ 𝑉 {𝑝 ∣ ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}} ∈ V)
18		abrexex2g 7959	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ ∀𝑎 ∈ 𝑉 {𝑝 ∣ ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}} ∈ V) → {𝑝 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}} ∈ V)
19	17, 18	mpdan 699	. 2 ⊢ (𝑉 ∈ 𝑊 → {𝑝 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}} ∈ V)
20	2, 7, 8, 19	fvmptd 6997	1 ⊢ (𝑉 ∈ 𝑊 → (Pairs‘𝑉) = {𝑝 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1569   ∈ wcel 2142  ∃!weu 2595  {cab 2740  ∀wral 3078  ∃wrex 3088  Vcvv 3454  {cpr 4590   ↦ cmpt 5191  ‘cfv 6536  Pairscspr 48254

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-iota 6492  df-fun 6538  df-fv 6544  df-spr 48255

This theorem is used by:  sprvalpw  48257  sprssspr  48258  prprspr2  48295