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Theorem scotteqd 42609
Description: Equality theorem for the Scott operation. (Contributed by Rohan Ridenour, 9-Aug-2023.)
Hypothesis
Ref Expression
scotteqd.1 (πœ‘ β†’ 𝐴 = 𝐡)
Assertion
Ref Expression
scotteqd (πœ‘ β†’ Scott 𝐴 = Scott 𝐡)

Proof of Theorem scotteqd
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 scotteqd.1 . . 3 (πœ‘ β†’ 𝐴 = 𝐡)
21adantr 482 . . . 4 ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐴 = 𝐡)
32raleqdv 3312 . . 3 ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ (βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦) ↔ βˆ€π‘¦ ∈ 𝐡 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)))
41, 3rabeqbidva 3422 . 2 (πœ‘ β†’ {π‘₯ ∈ 𝐴 ∣ βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)} = {π‘₯ ∈ 𝐡 ∣ βˆ€π‘¦ ∈ 𝐡 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)})
5 df-scott 42608 . 2 Scott 𝐴 = {π‘₯ ∈ 𝐴 ∣ βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)}
6 df-scott 42608 . 2 Scott 𝐡 = {π‘₯ ∈ 𝐡 ∣ βˆ€π‘¦ ∈ 𝐡 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)}
74, 5, 63eqtr4g 2798 1 (πœ‘ β†’ Scott 𝐴 = Scott 𝐡)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  {crab 3406   βŠ† wss 3914  β€˜cfv 6500  rankcrnk 9707  Scott cscott 42607
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 3062  df-rab 3407  df-scott 42608
This theorem is referenced by:  scotteq  42610  dfcoll2  42624  colleq12d  42625
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