Theorem scottab 44668

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 1916	. 2 ⊢ Ⅎ𝑥𝜓
2		scottab.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
3	1, 2	scottabf 44667	1 ⊢ Scott {𝑥 ∣ 𝜑} = {𝑥 ∣ (𝜑 ∧ ∀𝑦(𝜓 → (rank‘𝑥) ⊆ (rank‘𝑦)))}

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1540   = wceq 1542  {cab 2714   ⊆ wss 3889  ‘cfv 6498  rankcrnk 9687  Scott cscott 44662

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ral 3052  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-iota 6454  df-fv 6506  df-scott 44663

This theorem is referenced by: (None)