Theorem sprsymrelfolem1 47490

Description: Lemma 1 for sprsymrelfo 47495. (Contributed by AV, 22-Nov-2021.)

Hypothesis

Ref	Expression
sprsymrelfo.q	⊢ 𝑄 = {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)}

Assertion

Ref	Expression
sprsymrelfolem1	⊢ 𝑄 ∈ 𝒫 (Pairs‘𝑉)

Distinct variable group:   𝑉,𝑞

Allowed substitution hints:   𝑄(𝑞,𝑎,𝑏)   𝑅(𝑞,𝑎,𝑏)   𝑉(𝑎,𝑏)

Proof of Theorem sprsymrelfolem1

Step	Hyp	Ref	Expression
1		sprsymrelfo.q	. 2 ⊢ 𝑄 = {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)}
2		fvex 6871	. . 3 ⊢ (Pairs‘𝑉) ∈ V
3		ssrab2 4043	. . 3 ⊢ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)} ⊆ (Pairs‘𝑉)
4	2, 3	elpwi2 5290	. 2 ⊢ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)} ∈ 𝒫 (Pairs‘𝑉)
5	1, 4	eqeltri 2824	1 ⊢ 𝑄 ∈ 𝒫 (Pairs‘𝑉)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ∀wral 3044  {crab 3405  Vcvv 3447  𝒫 cpw 4563  {cpr 4591   class class class wbr 5107  ‘cfv 6511  Pairscspr 47475

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-ext 2701  ax-sep 5251  ax-nul 5261

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-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-pw 4565  df-sn 4590  df-pr 4592  df-uni 4872  df-iota 6464  df-fv 6519

This theorem is referenced by:  sprsymrelfo  47495