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Theorem sprsymrelfolem1 47502
Description: Lemma 1 for sprsymrelfo 47507. (Contributed by AV, 22-Nov-2021.)
Hypothesis
Ref Expression
sprsymrelfo.q 𝑄 = {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (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
StepHypRef Expression
1 sprsymrelfo.q . 2 𝑄 = {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)}
2 fvex 6830 . . 3 (Pairs‘𝑉) ∈ V
3 ssrab2 4028 . . 3 {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)} ⊆ (Pairs‘𝑉)
42, 3elpwi2 5271 . 2 {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)} ∈ 𝒫 (Pairs‘𝑉)
51, 4eqeltri 2825 1 𝑄 ∈ 𝒫 (Pairs‘𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2110  wral 3045  {crab 3393  Vcvv 3434  𝒫 cpw 4548  {cpr 4576   class class class wbr 5089  cfv 6477  Pairscspr 47487
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 2112  ax-9 2120  ax-ext 2702  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 2067  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-rab 3394  df-v 3436  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4282  df-pw 4550  df-sn 4575  df-pr 4577  df-uni 4858  df-iota 6433  df-fv 6485
This theorem is referenced by:  sprsymrelfo  47507
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