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Theorem scotteqd 40756
Description: Equality theorem for the Scott operation. (Contributed by Rohan Ridenour, 9-Aug-2023.)
Hypothesis
Ref Expression
scotteqd.1 (𝜑𝐴 = 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.
StepHypRef Expression
1 scotteqd.1 . . 3 (𝜑𝐴 = 𝐵)
21adantr 484 . . . 4 ((𝜑𝑥𝐴) → 𝐴 = 𝐵)
32raleqdv 3396 . . 3 ((𝜑𝑥𝐴) → (∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) ↔ ∀𝑦𝐵 (rank‘𝑥) ⊆ (rank‘𝑦)))
41, 3rabeqbidva 3463 . 2 (𝜑 → {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑥𝐵 ∣ ∀𝑦𝐵 (rank‘𝑥) ⊆ (rank‘𝑦)})
5 df-scott 40755 . 2 Scott 𝐴 = {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)}
6 df-scott 40755 . 2 Scott 𝐵 = {𝑥𝐵 ∣ ∀𝑦𝐵 (rank‘𝑥) ⊆ (rank‘𝑦)}
74, 5, 63eqtr4g 2881 1 (𝜑 → Scott 𝐴 = Scott 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  wral 3126  {crab 3130  wss 3910  cfv 6328  rankcrnk 9168  Scott cscott 40754
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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2793
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-ral 3131  df-rab 3135  df-scott 40755
This theorem is referenced by:  scotteq  40757  dfcoll2  40771  colleq12d  40772
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