Theorem scottab 44253

Description: Value of the Scott operation at a class abstraction. (Contributed by Rohan Ridenour, 14-Aug-2023.)

Hypothesis

Ref	Expression
scottab.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
scottab	⊢ Scott {𝑥 ∣ 𝜑} = {𝑥 ∣ (𝜑 ∧ ∀𝑦(𝜓 → (rank‘𝑥) ⊆ (rank‘𝑦)))}

Distinct variable groups:   𝑥,𝑦   𝜑,𝑦   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem scottab

Step	Hyp	Ref	Expression
1		nfv 1915	. 2 ⊢ Ⅎ𝑥𝜓
2		scottab.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
3	1, 2	scottabf 44252	1 ⊢ Scott {𝑥 ∣ 𝜑} = {𝑥 ∣ (𝜑 ∧ ∀𝑦(𝜓 → (rank‘𝑥) ⊆ (rank‘𝑦)))}

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1539   = wceq 1541  {cab 2708   ⊆ wss 3900  ‘cfv 6477  rankcrnk 9648  Scott cscott 44247

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-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702

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-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ral 3046  df-rab 3394  df-v 3436  df-dif 3903  df-un 3905  df-ss 3917  df-nul 4282  df-if 4474  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-br 5090  df-iota 6433  df-fv 6485  df-scott 44248

This theorem is referenced by: (None)