Theorem sprvalpw 47087

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

Assertion

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

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

Proof of Theorem sprvalpw

Step	Hyp	Ref	Expression
1		sprval 47086	. 2 ⊢ (𝑉 ∈ 𝑊 → (Pairs‘𝑉) = {𝑝 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}})
2		prssi 4822	. . . . . . . 8 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → {𝑎, 𝑏} ⊆ 𝑉)
3		eleq1 2814	. . . . . . . . 9 ⊢ (𝑝 = {𝑎, 𝑏} → (𝑝 ∈ 𝒫 𝑉 ↔ {𝑎, 𝑏} ∈ 𝒫 𝑉))
4		prex 5430	. . . . . . . . . 10 ⊢ {𝑎, 𝑏} ∈ V
5	4	elpw 4603	. . . . . . . . 9 ⊢ ({𝑎, 𝑏} ∈ 𝒫 𝑉 ↔ {𝑎, 𝑏} ⊆ 𝑉)
6	3, 5	bitrdi 286	. . . . . . . 8 ⊢ (𝑝 = {𝑎, 𝑏} → (𝑝 ∈ 𝒫 𝑉 ↔ {𝑎, 𝑏} ⊆ 𝑉))
7	2, 6	syl5ibrcom 246	. . . . . . 7 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝑝 = {𝑎, 𝑏} → 𝑝 ∈ 𝒫 𝑉))
8	7	rexlimivv 3190	. . . . . 6 ⊢ (∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏} → 𝑝 ∈ 𝒫 𝑉)
9	8	pm4.71ri 559	. . . . 5 ⊢ (∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏} ↔ (𝑝 ∈ 𝒫 𝑉 ∧ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}))
10	9	a1i 11	. . . 4 ⊢ (𝑉 ∈ 𝑊 → (∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏} ↔ (𝑝 ∈ 𝒫 𝑉 ∧ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏})))
11	10	abbidv 2795	. . 3 ⊢ (𝑉 ∈ 𝑊 → {𝑝 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}} = {𝑝 ∣ (𝑝 ∈ 𝒫 𝑉 ∧ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏})})
12		df-rab 3421	. . 3 ⊢ {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}} = {𝑝 ∣ (𝑝 ∈ 𝒫 𝑉 ∧ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏})}
13	11, 12	eqtr4di 2784	. 2 ⊢ (𝑉 ∈ 𝑊 → {𝑝 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}} = {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}})
14	1, 13	eqtrd 2766	1 ⊢ (𝑉 ∈ 𝑊 → (Pairs‘𝑉) = {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}})

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1534   ∈ wcel 2099  {cab 2703  ∃wrex 3060  {crab 3420   ⊆ wss 3948  𝒫 cpw 4599  {cpr 4627  ‘cfv 6545  Pairscspr 47084

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pr 5425  ax-un 7737

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ral 3052  df-rex 3061  df-rab 3421  df-v 3466  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4325  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4908  df-iun 4997  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-iota 6497  df-fun 6547  df-fv 6553  df-spr 47085

This theorem is referenced by:  sprvalpwn0  47090  sprel  47091  prelspr  47093