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Theorem scotteqd 42929
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 482 . . . 4 ((𝜑𝑥𝐴) → 𝐴 = 𝐵)
32raleqdv 3326 . . 3 ((𝜑𝑥𝐴) → (∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) ↔ ∀𝑦𝐵 (rank‘𝑥) ⊆ (rank‘𝑦)))
41, 3rabeqbidva 3449 . 2 (𝜑 → {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑥𝐵 ∣ ∀𝑦𝐵 (rank‘𝑥) ⊆ (rank‘𝑦)})
5 df-scott 42928 . 2 Scott 𝐴 = {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)}
6 df-scott 42928 . 2 Scott 𝐵 = {𝑥𝐵 ∣ ∀𝑦𝐵 (rank‘𝑥) ⊆ (rank‘𝑦)}
74, 5, 63eqtr4g 2798 1 (𝜑 → Scott 𝐴 = Scott 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3062  {crab 3433  wss 3947  cfv 6540  rankcrnk 9754  Scott cscott 42927
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-scott 42928
This theorem is referenced by:  scotteq  42930  dfcoll2  42944  colleq12d  42945
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