Theorem sprval 44819

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 44818	. . 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 3334	. . . . 5 ⊢ (𝑣 = 𝑉 → (∃𝑏 ∈ 𝑣 𝑝 = {𝑎, 𝑏} ↔ ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}))
5	3, 4	rexeqbidv 3328	. . . 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 2808	. 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑣 = 𝑉) → {𝑝 ∣ ∃𝑎 ∈ 𝑣 ∃𝑏 ∈ 𝑣 𝑝 = {𝑎, 𝑏}} = {𝑝 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}})
8		elex 3440	. 2 ⊢ (𝑉 ∈ 𝑊 → 𝑉 ∈ V)
9		zfpair2 5348	. . . . . . . 8 ⊢ {𝑎, 𝑏} ∈ V
10		eueq 3638	. . . . . . . 8 ⊢ ({𝑎, 𝑏} ∈ V ↔ ∃!𝑝 𝑝 = {𝑎, 𝑏})
11	9, 10	mpbi 229	. . . . . . 7 ⊢ ∃!𝑝 𝑝 = {𝑎, 𝑏}
12		euabex 5370	. . . . . . 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 3108	. . . . 5 ⊢ (𝑉 ∈ 𝑊 → ∀𝑏 ∈ 𝑉 {𝑝 ∣ 𝑝 = {𝑎, 𝑏}} ∈ V)
15		abrexex2g 7780	. . . . 5 ⊢ ((𝑉 ∈ 𝑊 ∧ ∀𝑏 ∈ 𝑉 {𝑝 ∣ 𝑝 = {𝑎, 𝑏}} ∈ V) → {𝑝 ∣ ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}} ∈ V)
16	14, 15	mpdan 683	. . . 4 ⊢ (𝑉 ∈ 𝑊 → {𝑝 ∣ ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}} ∈ V)
17	16	ralrimivw 3108	. . 3 ⊢ (𝑉 ∈ 𝑊 → ∀𝑎 ∈ 𝑉 {𝑝 ∣ ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}} ∈ V)
18		abrexex2g 7780	. . 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 683	. 2 ⊢ (𝑉 ∈ 𝑊 → {𝑝 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}} ∈ V)
20	2, 7, 8, 19	fvmptd 6864	1 ⊢ (𝑉 ∈ 𝑊 → (Pairs‘𝑉) = {𝑝 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}})

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∃!weu 2568  {cab 2715  ∀wral 3063  ∃wrex 3064  Vcvv 3422  {cpr 4560   ↦ cmpt 5153  ‘cfv 6418  Pairscspr 44817

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-spr 44818

This theorem is referenced by:  sprvalpw  44820  sprssspr  44821  prprspr2  44858