Theorem scotteqd 44818

Description: Equality theorem for the Scott operation. (Contributed by Rohan Ridenour, 9-Aug-2023.)

Hypothesis

Ref	Expression
scotteqd.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
scotteqd	⊢ (𝜑 → Scott 𝐴 = Scott 𝐵)

Proof of Theorem scotteqd

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		scotteqd.1	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
2	1	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐴 = 𝐵)
3	2	raleqdv 3322	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) ↔ ∀𝑦 ∈ 𝐵 (rank‘𝑥) ⊆ (rank‘𝑦)))
4	1, 3	rabeqbidva 3432	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 (rank‘𝑥) ⊆ (rank‘𝑦)})
5		df-scott 44817	. 2 ⊢ Scott 𝐴 = {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)}
6		df-scott 44817	. 2 ⊢ Scott 𝐵 = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 (rank‘𝑥) ⊆ (rank‘𝑦)}
7	4, 5, 6	3eqtr4g 2824	1 ⊢ (𝜑 → Scott 𝐴 = Scott 𝐵)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1562   ∈ wcel 2144  ∀wral 3078  {crab 3416   ⊆ wss 3906  ‘cfv 6523  rankcrnk 9723  Scott cscott 44816

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ral 3079  df-rex 3089  df-rab 3417  df-scott 44817

This theorem is referenced by:  scotteq  44819  dfcoll2  44833  colleq12d  44834