Theorem sprval 47480

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 47479	. . 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 22	. . . . 5 ⊢ (𝑣 = 𝑉 → 𝑣 = 𝑉)
4		rexeq 3295	. . . . 5 ⊢ (𝑣 = 𝑉 → (∃𝑏 ∈ 𝑣 𝑝 = {𝑎, 𝑏} ↔ ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}))
5	3, 4	rexeqbidv 3320	. . . 4 ⊢ (𝑣 = 𝑉 → (∃𝑎 ∈ 𝑣 ∃𝑏 ∈ 𝑣 𝑝 = {𝑎, 𝑏} ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}))
6	5	adantl 481	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑣 = 𝑉) → (∃𝑎 ∈ 𝑣 ∃𝑏 ∈ 𝑣 𝑝 = {𝑎, 𝑏} ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}))
7	6	abbidv 2795	. 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑣 = 𝑉) → {𝑝 ∣ ∃𝑎 ∈ 𝑣 ∃𝑏 ∈ 𝑣 𝑝 = {𝑎, 𝑏}} = {𝑝 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}})
8		elex 3468	. 2 ⊢ (𝑉 ∈ 𝑊 → 𝑉 ∈ V)
9		zfpair2 5388	. . . . . . . 8 ⊢ {𝑎, 𝑏} ∈ V
10		eueq 3679	. . . . . . . 8 ⊢ ({𝑎, 𝑏} ∈ V ↔ ∃!𝑝 𝑝 = {𝑎, 𝑏})
11	9, 10	mpbi 230	. . . . . . 7 ⊢ ∃!𝑝 𝑝 = {𝑎, 𝑏}
12		euabex 5421	. . . . . . 7 ⊢ (∃!𝑝 𝑝 = {𝑎, 𝑏} → {𝑝 ∣ 𝑝 = {𝑎, 𝑏}} ∈ V)
13	11, 12	mp1i 13	. . . . . 6 ⊢ (𝑉 ∈ 𝑊 → {𝑝 ∣ 𝑝 = {𝑎, 𝑏}} ∈ V)
14	13	ralrimivw 3129	. . . . 5 ⊢ (𝑉 ∈ 𝑊 → ∀𝑏 ∈ 𝑉 {𝑝 ∣ 𝑝 = {𝑎, 𝑏}} ∈ V)
15		abrexex2g 7943	. . . . 5 ⊢ ((𝑉 ∈ 𝑊 ∧ ∀𝑏 ∈ 𝑉 {𝑝 ∣ 𝑝 = {𝑎, 𝑏}} ∈ V) → {𝑝 ∣ ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}} ∈ V)
16	14, 15	mpdan 687	. . . 4 ⊢ (𝑉 ∈ 𝑊 → {𝑝 ∣ ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}} ∈ V)
17	16	ralrimivw 3129	. . 3 ⊢ (𝑉 ∈ 𝑊 → ∀𝑎 ∈ 𝑉 {𝑝 ∣ ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}} ∈ V)
18		abrexex2g 7943	. . 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 687	. 2 ⊢ (𝑉 ∈ 𝑊 → {𝑝 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}} ∈ V)
20	2, 7, 8, 19	fvmptd 6975	1 ⊢ (𝑉 ∈ 𝑊 → (Pairs‘𝑉) = {𝑝 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃!weu 2561  {cab 2707  ∀wral 3044  ∃wrex 3053  Vcvv 3447  {cpr 4591   ↦ cmpt 5188  ‘cfv 6511  Pairscspr 47478

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6464  df-fun 6513  df-fv 6519  df-spr 47479

This theorem is referenced by:  sprvalpw  47481  sprssspr  47482  prprspr2  47519