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Theorem scottabf 42612
Description: Value of the Scott operation at a class abstraction. Variant of scottab 42613 with a nonfreeness hypothesis instead of a disjoint variable condition. (Contributed by Rohan Ridenour, 14-Aug-2023.)
Hypotheses
Ref Expression
scottabf.1 β„²π‘₯πœ“
scottabf.2 (π‘₯ = 𝑦 β†’ (πœ‘ ↔ πœ“))
Assertion
Ref Expression
scottabf Scott {π‘₯ ∣ πœ‘} = {π‘₯ ∣ (πœ‘ ∧ βˆ€π‘¦(πœ“ β†’ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)))}
Distinct variable groups:   πœ‘,𝑦   π‘₯,𝑦
Allowed substitution hints:   πœ‘(π‘₯)   πœ“(π‘₯,𝑦)

Proof of Theorem scottabf
Dummy variables 𝑧 𝑀 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-scott 42608 . 2 Scott {π‘₯ ∣ πœ‘} = {𝑧 ∈ {π‘₯ ∣ πœ‘} ∣ βˆ€π‘€ ∈ {π‘₯ ∣ πœ‘} (rankβ€˜π‘§) βŠ† (rankβ€˜π‘€)}
2 df-rab 3407 . 2 {𝑧 ∈ {π‘₯ ∣ πœ‘} ∣ βˆ€π‘€ ∈ {π‘₯ ∣ πœ‘} (rankβ€˜π‘§) βŠ† (rankβ€˜π‘€)} = {𝑧 ∣ (𝑧 ∈ {π‘₯ ∣ πœ‘} ∧ βˆ€π‘€ ∈ {π‘₯ ∣ πœ‘} (rankβ€˜π‘§) βŠ† (rankβ€˜π‘€))}
3 eqabc 2876 . . 3 ({𝑧 ∣ (𝑧 ∈ {π‘₯ ∣ πœ‘} ∧ βˆ€π‘€ ∈ {π‘₯ ∣ πœ‘} (rankβ€˜π‘§) βŠ† (rankβ€˜π‘€))} = {π‘₯ ∣ (πœ‘ ∧ βˆ€π‘¦(πœ“ β†’ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)))} ↔ βˆ€π‘§((𝑧 ∈ {π‘₯ ∣ πœ‘} ∧ βˆ€π‘€ ∈ {π‘₯ ∣ πœ‘} (rankβ€˜π‘§) βŠ† (rankβ€˜π‘€)) ↔ 𝑧 ∈ {π‘₯ ∣ (πœ‘ ∧ βˆ€π‘¦(πœ“ β†’ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)))}))
4 nfsab1 2718 . . . . . 6 β„²π‘₯ 𝑧 ∈ {π‘₯ ∣ πœ‘}
5 nfab1 2906 . . . . . . 7 β„²π‘₯{π‘₯ ∣ πœ‘}
6 nfv 1918 . . . . . . 7 β„²π‘₯(rankβ€˜π‘§) βŠ† (rankβ€˜π‘€)
75, 6nfralw 3293 . . . . . 6 β„²π‘₯βˆ€π‘€ ∈ {π‘₯ ∣ πœ‘} (rankβ€˜π‘§) βŠ† (rankβ€˜π‘€)
84, 7nfan 1903 . . . . 5 β„²π‘₯(𝑧 ∈ {π‘₯ ∣ πœ‘} ∧ βˆ€π‘€ ∈ {π‘₯ ∣ πœ‘} (rankβ€˜π‘§) βŠ† (rankβ€˜π‘€))
9 vex 3451 . . . . 5 𝑧 ∈ V
10 abid 2714 . . . . . . 7 (π‘₯ ∈ {π‘₯ ∣ πœ‘} ↔ πœ‘)
11 eleq1w 2817 . . . . . . 7 (π‘₯ = 𝑧 β†’ (π‘₯ ∈ {π‘₯ ∣ πœ‘} ↔ 𝑧 ∈ {π‘₯ ∣ πœ‘}))
1210, 11bitr3id 285 . . . . . 6 (π‘₯ = 𝑧 β†’ (πœ‘ ↔ 𝑧 ∈ {π‘₯ ∣ πœ‘}))
13 df-clab 2711 . . . . . . . . . . . 12 (𝑦 ∈ {π‘₯ ∣ πœ‘} ↔ [𝑦 / π‘₯]πœ‘)
14 scottabf.1 . . . . . . . . . . . . 13 β„²π‘₯πœ“
15 scottabf.2 . . . . . . . . . . . . 13 (π‘₯ = 𝑦 β†’ (πœ‘ ↔ πœ“))
1614, 15sbiev 2309 . . . . . . . . . . . 12 ([𝑦 / π‘₯]πœ‘ ↔ πœ“)
1713, 16bitr2i 276 . . . . . . . . . . 11 (πœ“ ↔ 𝑦 ∈ {π‘₯ ∣ πœ‘})
18 eleq1w 2817 . . . . . . . . . . 11 (𝑦 = 𝑀 β†’ (𝑦 ∈ {π‘₯ ∣ πœ‘} ↔ 𝑀 ∈ {π‘₯ ∣ πœ‘}))
1917, 18bitrid 283 . . . . . . . . . 10 (𝑦 = 𝑀 β†’ (πœ“ ↔ 𝑀 ∈ {π‘₯ ∣ πœ‘}))
2019adantl 483 . . . . . . . . 9 ((π‘₯ = 𝑧 ∧ 𝑦 = 𝑀) β†’ (πœ“ ↔ 𝑀 ∈ {π‘₯ ∣ πœ‘}))
21 simpl 484 . . . . . . . . . . 11 ((π‘₯ = 𝑧 ∧ 𝑦 = 𝑀) β†’ π‘₯ = 𝑧)
2221fveq2d 6850 . . . . . . . . . 10 ((π‘₯ = 𝑧 ∧ 𝑦 = 𝑀) β†’ (rankβ€˜π‘₯) = (rankβ€˜π‘§))
23 simpr 486 . . . . . . . . . . 11 ((π‘₯ = 𝑧 ∧ 𝑦 = 𝑀) β†’ 𝑦 = 𝑀)
2423fveq2d 6850 . . . . . . . . . 10 ((π‘₯ = 𝑧 ∧ 𝑦 = 𝑀) β†’ (rankβ€˜π‘¦) = (rankβ€˜π‘€))
2522, 24sseq12d 3981 . . . . . . . . 9 ((π‘₯ = 𝑧 ∧ 𝑦 = 𝑀) β†’ ((rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦) ↔ (rankβ€˜π‘§) βŠ† (rankβ€˜π‘€)))
2620, 25imbi12d 345 . . . . . . . 8 ((π‘₯ = 𝑧 ∧ 𝑦 = 𝑀) β†’ ((πœ“ β†’ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)) ↔ (𝑀 ∈ {π‘₯ ∣ πœ‘} β†’ (rankβ€˜π‘§) βŠ† (rankβ€˜π‘€))))
2726cbvaldvaw 2042 . . . . . . 7 (π‘₯ = 𝑧 β†’ (βˆ€π‘¦(πœ“ β†’ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)) ↔ βˆ€π‘€(𝑀 ∈ {π‘₯ ∣ πœ‘} β†’ (rankβ€˜π‘§) βŠ† (rankβ€˜π‘€))))
28 df-ral 3062 . . . . . . 7 (βˆ€π‘€ ∈ {π‘₯ ∣ πœ‘} (rankβ€˜π‘§) βŠ† (rankβ€˜π‘€) ↔ βˆ€π‘€(𝑀 ∈ {π‘₯ ∣ πœ‘} β†’ (rankβ€˜π‘§) βŠ† (rankβ€˜π‘€)))
2927, 28bitr4di 289 . . . . . 6 (π‘₯ = 𝑧 β†’ (βˆ€π‘¦(πœ“ β†’ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)) ↔ βˆ€π‘€ ∈ {π‘₯ ∣ πœ‘} (rankβ€˜π‘§) βŠ† (rankβ€˜π‘€)))
3012, 29anbi12d 632 . . . . 5 (π‘₯ = 𝑧 β†’ ((πœ‘ ∧ βˆ€π‘¦(πœ“ β†’ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦))) ↔ (𝑧 ∈ {π‘₯ ∣ πœ‘} ∧ βˆ€π‘€ ∈ {π‘₯ ∣ πœ‘} (rankβ€˜π‘§) βŠ† (rankβ€˜π‘€))))
318, 9, 30elabf 3631 . . . 4 (𝑧 ∈ {π‘₯ ∣ (πœ‘ ∧ βˆ€π‘¦(πœ“ β†’ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)))} ↔ (𝑧 ∈ {π‘₯ ∣ πœ‘} ∧ βˆ€π‘€ ∈ {π‘₯ ∣ πœ‘} (rankβ€˜π‘§) βŠ† (rankβ€˜π‘€)))
3231bicomi 223 . . 3 ((𝑧 ∈ {π‘₯ ∣ πœ‘} ∧ βˆ€π‘€ ∈ {π‘₯ ∣ πœ‘} (rankβ€˜π‘§) βŠ† (rankβ€˜π‘€)) ↔ 𝑧 ∈ {π‘₯ ∣ (πœ‘ ∧ βˆ€π‘¦(πœ“ β†’ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)))})
333, 32mpgbir 1802 . 2 {𝑧 ∣ (𝑧 ∈ {π‘₯ ∣ πœ‘} ∧ βˆ€π‘€ ∈ {π‘₯ ∣ πœ‘} (rankβ€˜π‘§) βŠ† (rankβ€˜π‘€))} = {π‘₯ ∣ (πœ‘ ∧ βˆ€π‘¦(πœ“ β†’ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)))}
341, 2, 333eqtri 2765 1 Scott {π‘₯ ∣ πœ‘} = {π‘₯ ∣ (πœ‘ ∧ βˆ€π‘¦(πœ“ β†’ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)))}
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397  βˆ€wal 1540   = wceq 1542  β„²wnf 1786  [wsb 2068   ∈ wcel 2107  {cab 2710  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-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-iota 6452  df-fv 6508  df-scott 42608
This theorem is referenced by:  scottab  42613  scottabes  42614
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