Theorem sprsymrelfolem1 47417

Description: Lemma 1 for sprsymrelfo 47422. (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 6920	. . 3 ⊢ (Pairs‘𝑉) ∈ V
3		ssrab2 4090	. . 3 ⊢ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)} ⊆ (Pairs‘𝑉)
4	2, 3	elpwi2 5341	. 2 ⊢ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)} ∈ 𝒫 (Pairs‘𝑉)
5	1, 4	eqeltri 2835	1 ⊢ 𝑄 ∈ 𝒫 (Pairs‘𝑉)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2106  ∀wral 3059  {crab 3433  Vcvv 3478  𝒫 cpw 4605  {cpr 4633   class class class wbr 5148  ‘cfv 6563  Pairscspr 47402

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-pw 4607  df-sn 4632  df-pr 4634  df-uni 4913  df-iota 6516  df-fv 6571

This theorem is referenced by:  sprsymrelfo  47422