Theorem scotteqd 41744

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 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐴 = 𝐵)
3	2	raleqdv 3339	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) ↔ ∀𝑦 ∈ 𝐵 (rank‘𝑥) ⊆ (rank‘𝑦)))
4	1, 3	rabeqbidva 3411	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 (rank‘𝑥) ⊆ (rank‘𝑦)})
5		df-scott 41743	. 2 ⊢ Scott 𝐴 = {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)}
6		df-scott 41743	. 2 ⊢ Scott 𝐵 = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 (rank‘𝑥) ⊆ (rank‘𝑦)}
7	4, 5, 6	3eqtr4g 2804	1 ⊢ (𝜑 → Scott 𝐴 = Scott 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  {crab 3067   ⊆ wss 3883  ‘cfv 6418  rankcrnk 9452  Scott cscott 41742

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rab 3072  df-scott 41743

This theorem is referenced by:  scotteq  41745  dfcoll2  41759  colleq12d  41760