Theorem scotteqd 44696

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 482	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐴 = 𝐵)
3	2	raleqdv 3299	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) ↔ ∀𝑦 ∈ 𝐵 (rank‘𝑥) ⊆ (rank‘𝑦)))
4	1, 3	rabeqbidva 3409	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 (rank‘𝑥) ⊆ (rank‘𝑦)})
5		df-scott 44695	. 2 ⊢ Scott 𝐴 = {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)}
6		df-scott 44695	. 2 ⊢ Scott 𝐵 = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 (rank‘𝑥) ⊆ (rank‘𝑦)}
7	4, 5, 6	3eqtr4g 2801	1 ⊢ (𝜑 → Scott 𝐴 = Scott 𝐵)

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1548   ∈ wcel 2121  ∀wral 3055  {crab 3393   ⊆ wss 3885  ‘cfv 6489  rankcrnk 9682  Scott cscott 44694

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ral 3056  df-rex 3066  df-rab 3394  df-scott 44695

This theorem is referenced by:  scotteq  44697  dfcoll2  44711  colleq12d  44712