Theorem scotteqd 42986

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 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐴 = 𝐵)
3	2	raleqdv 3325	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) ↔ ∀𝑦 ∈ 𝐵 (rank‘𝑥) ⊆ (rank‘𝑦)))
4	1, 3	rabeqbidva 3448	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 (rank‘𝑥) ⊆ (rank‘𝑦)})
5		df-scott 42985	. 2 ⊢ Scott 𝐴 = {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)}
6		df-scott 42985	. 2 ⊢ Scott 𝐵 = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 (rank‘𝑥) ⊆ (rank‘𝑦)}
7	4, 5, 6	3eqtr4g 2797	1 ⊢ (𝜑 → Scott 𝐴 = Scott 𝐵)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3061  {crab 3432   ⊆ wss 3948  ‘cfv 6543  rankcrnk 9757  Scott cscott 42984

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-scott 42985

This theorem is referenced by:  scotteq  42987  dfcoll2  43001  colleq12d  43002