Theorem sprval 47721

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 47720	. . 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 3292	. . . . 5 ⊢ (𝑣 = 𝑉 → (∃𝑏 ∈ 𝑣 𝑝 = {𝑎, 𝑏} ↔ ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}))
5	3, 4	rexeqbidv 3317	. . . 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 2802	. 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑣 = 𝑉) → {𝑝 ∣ ∃𝑎 ∈ 𝑣 ∃𝑏 ∈ 𝑣 𝑝 = {𝑎, 𝑏}} = {𝑝 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}})
8		elex 3461	. 2 ⊢ (𝑉 ∈ 𝑊 → 𝑉 ∈ V)
9		zfpair2 5378	. . . . . . . 8 ⊢ {𝑎, 𝑏} ∈ V
10		eueq 3666	. . . . . . . 8 ⊢ ({𝑎, 𝑏} ∈ V ↔ ∃!𝑝 𝑝 = {𝑎, 𝑏})
11	9, 10	mpbi 230	. . . . . . 7 ⊢ ∃!𝑝 𝑝 = {𝑎, 𝑏}
12		euabex 5409	. . . . . . 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 3132	. . . . 5 ⊢ (𝑉 ∈ 𝑊 → ∀𝑏 ∈ 𝑉 {𝑝 ∣ 𝑝 = {𝑎, 𝑏}} ∈ V)
15		abrexex2g 7908	. . . . 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 3132	. . 3 ⊢ (𝑉 ∈ 𝑊 → ∀𝑎 ∈ 𝑉 {𝑝 ∣ ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}} ∈ V)
18		abrexex2g 7908	. . 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 6948	1 ⊢ (𝑉 ∈ 𝑊 → (Pairs‘𝑉) = {𝑝 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∃!weu 2568  {cab 2714  ∀wral 3051  ∃wrex 3060  Vcvv 3440  {cpr 4582   ↦ cmpt 5179  ‘cfv 6492  Pairscspr 47719

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fv 6500  df-spr 47720

This theorem is referenced by:  sprvalpw  47722  sprssspr  47723  prprspr2  47760