Theorem sprsymrelfolem1 48098

Description: Lemma 1 for sprsymrelfo 48103. (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 6880	. . 3 ⊢ (Pairs‘𝑉) ∈ V
3		ssrab2 4033	. . 3 ⊢ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)} ⊆ (Pairs‘𝑉)
4	2, 3	elpwi2 5291	. 2 ⊢ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)} ∈ 𝒫 (Pairs‘𝑉)
5	1, 4	eqeltri 2858	1 ⊢ 𝑄 ∈ 𝒫 (Pairs‘𝑉)

Syntax hints:   → wi 4   = wceq 1560   ∈ wcel 2142  ∀wral 3076  {crab 3414  Vcvv 3454  𝒫 cpw 4555  {cpr 4584   class class class wbr 5100  ‘cfv 6521  Pairscspr 48083

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-nul 5256

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-pw 4557  df-sn 4583  df-pr 4585  df-uni 4866  df-iota 6477  df-fv 6529

This theorem is referenced by:  sprsymrelfo  48103