Theorem sprsymrelfolem1 47602

Description: Lemma 1 for sprsymrelfo 47607. (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 6835	. . 3 ⊢ (Pairs‘𝑉) ∈ V
3		ssrab2 4027	. . 3 ⊢ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)} ⊆ (Pairs‘𝑉)
4	2, 3	elpwi2 5271	. 2 ⊢ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)} ∈ 𝒫 (Pairs‘𝑉)
5	1, 4	eqeltri 2827	1 ⊢ 𝑄 ∈ 𝒫 (Pairs‘𝑉)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  ∀wral 3047  {crab 3395  Vcvv 3436  𝒫 cpw 4547  {cpr 4575   class class class wbr 5089  ‘cfv 6481  Pairscspr 47587

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 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242

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-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-pw 4549  df-sn 4574  df-pr 4576  df-uni 4857  df-iota 6437  df-fv 6489

This theorem is referenced by:  sprsymrelfo  47607