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Theorem scottabf 40991
 Description: Value of the Scott operation at a class abstraction. Variant of scottab 40992 with a nonfreeness hypothesis instead of a disjoint variable condition. (Contributed by Rohan Ridenour, 14-Aug-2023.)
Hypotheses
Ref Expression
scottabf.1 𝑥𝜓
scottabf.2 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
scottabf Scott {𝑥𝜑} = {𝑥 ∣ (𝜑 ∧ ∀𝑦(𝜓 → (rank‘𝑥) ⊆ (rank‘𝑦)))}
Distinct variable groups:   𝜑,𝑦   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥,𝑦)

Proof of Theorem scottabf
Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-scott 40987 . 2 Scott {𝑥𝜑} = {𝑧 ∈ {𝑥𝜑} ∣ ∀𝑤 ∈ {𝑥𝜑} (rank‘𝑧) ⊆ (rank‘𝑤)}
2 df-rab 3115 . 2 {𝑧 ∈ {𝑥𝜑} ∣ ∀𝑤 ∈ {𝑥𝜑} (rank‘𝑧) ⊆ (rank‘𝑤)} = {𝑧 ∣ (𝑧 ∈ {𝑥𝜑} ∧ ∀𝑤 ∈ {𝑥𝜑} (rank‘𝑧) ⊆ (rank‘𝑤))}
3 abeq1 2923 . . 3 ({𝑧 ∣ (𝑧 ∈ {𝑥𝜑} ∧ ∀𝑤 ∈ {𝑥𝜑} (rank‘𝑧) ⊆ (rank‘𝑤))} = {𝑥 ∣ (𝜑 ∧ ∀𝑦(𝜓 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ↔ ∀𝑧((𝑧 ∈ {𝑥𝜑} ∧ ∀𝑤 ∈ {𝑥𝜑} (rank‘𝑧) ⊆ (rank‘𝑤)) ↔ 𝑧 ∈ {𝑥 ∣ (𝜑 ∧ ∀𝑦(𝜓 → (rank‘𝑥) ⊆ (rank‘𝑦)))}))
4 nfsab1 2785 . . . . . 6 𝑥 𝑧 ∈ {𝑥𝜑}
5 nfab1 2957 . . . . . . 7 𝑥{𝑥𝜑}
6 nfv 1915 . . . . . . 7 𝑥(rank‘𝑧) ⊆ (rank‘𝑤)
75, 6nfralw 3189 . . . . . 6 𝑥𝑤 ∈ {𝑥𝜑} (rank‘𝑧) ⊆ (rank‘𝑤)
84, 7nfan 1900 . . . . 5 𝑥(𝑧 ∈ {𝑥𝜑} ∧ ∀𝑤 ∈ {𝑥𝜑} (rank‘𝑧) ⊆ (rank‘𝑤))
9 vex 3444 . . . . 5 𝑧 ∈ V
10 abid 2780 . . . . . . 7 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
11 eleq1w 2872 . . . . . . 7 (𝑥 = 𝑧 → (𝑥 ∈ {𝑥𝜑} ↔ 𝑧 ∈ {𝑥𝜑}))
1210, 11bitr3id 288 . . . . . 6 (𝑥 = 𝑧 → (𝜑𝑧 ∈ {𝑥𝜑}))
13 df-clab 2777 . . . . . . . . . . . 12 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
14 scottabf.1 . . . . . . . . . . . . 13 𝑥𝜓
15 scottabf.2 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝜑𝜓))
1614, 15sbiev 2322 . . . . . . . . . . . 12 ([𝑦 / 𝑥]𝜑𝜓)
1713, 16bitr2i 279 . . . . . . . . . . 11 (𝜓𝑦 ∈ {𝑥𝜑})
18 eleq1w 2872 . . . . . . . . . . 11 (𝑦 = 𝑤 → (𝑦 ∈ {𝑥𝜑} ↔ 𝑤 ∈ {𝑥𝜑}))
1917, 18syl5bb 286 . . . . . . . . . 10 (𝑦 = 𝑤 → (𝜓𝑤 ∈ {𝑥𝜑}))
2019adantl 485 . . . . . . . . 9 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜓𝑤 ∈ {𝑥𝜑}))
21 simpl 486 . . . . . . . . . . 11 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝑥 = 𝑧)
2221fveq2d 6650 . . . . . . . . . 10 ((𝑥 = 𝑧𝑦 = 𝑤) → (rank‘𝑥) = (rank‘𝑧))
23 simpr 488 . . . . . . . . . . 11 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝑦 = 𝑤)
2423fveq2d 6650 . . . . . . . . . 10 ((𝑥 = 𝑧𝑦 = 𝑤) → (rank‘𝑦) = (rank‘𝑤))
2522, 24sseq12d 3948 . . . . . . . . 9 ((𝑥 = 𝑧𝑦 = 𝑤) → ((rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝑧) ⊆ (rank‘𝑤)))
2620, 25imbi12d 348 . . . . . . . 8 ((𝑥 = 𝑧𝑦 = 𝑤) → ((𝜓 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ (𝑤 ∈ {𝑥𝜑} → (rank‘𝑧) ⊆ (rank‘𝑤))))
2726cbvaldvaw 2045 . . . . . . 7 (𝑥 = 𝑧 → (∀𝑦(𝜓 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ ∀𝑤(𝑤 ∈ {𝑥𝜑} → (rank‘𝑧) ⊆ (rank‘𝑤))))
28 df-ral 3111 . . . . . . 7 (∀𝑤 ∈ {𝑥𝜑} (rank‘𝑧) ⊆ (rank‘𝑤) ↔ ∀𝑤(𝑤 ∈ {𝑥𝜑} → (rank‘𝑧) ⊆ (rank‘𝑤)))
2927, 28bitr4di 292 . . . . . 6 (𝑥 = 𝑧 → (∀𝑦(𝜓 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ ∀𝑤 ∈ {𝑥𝜑} (rank‘𝑧) ⊆ (rank‘𝑤)))
3012, 29anbi12d 633 . . . . 5 (𝑥 = 𝑧 → ((𝜑 ∧ ∀𝑦(𝜓 → (rank‘𝑥) ⊆ (rank‘𝑦))) ↔ (𝑧 ∈ {𝑥𝜑} ∧ ∀𝑤 ∈ {𝑥𝜑} (rank‘𝑧) ⊆ (rank‘𝑤))))
318, 9, 30elabf 3611 . . . 4 (𝑧 ∈ {𝑥 ∣ (𝜑 ∧ ∀𝑦(𝜓 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ↔ (𝑧 ∈ {𝑥𝜑} ∧ ∀𝑤 ∈ {𝑥𝜑} (rank‘𝑧) ⊆ (rank‘𝑤)))
3231bicomi 227 . . 3 ((𝑧 ∈ {𝑥𝜑} ∧ ∀𝑤 ∈ {𝑥𝜑} (rank‘𝑧) ⊆ (rank‘𝑤)) ↔ 𝑧 ∈ {𝑥 ∣ (𝜑 ∧ ∀𝑦(𝜓 → (rank‘𝑥) ⊆ (rank‘𝑦)))})
333, 32mpgbir 1801 . 2 {𝑧 ∣ (𝑧 ∈ {𝑥𝜑} ∧ ∀𝑤 ∈ {𝑥𝜑} (rank‘𝑧) ⊆ (rank‘𝑤))} = {𝑥 ∣ (𝜑 ∧ ∀𝑦(𝜓 → (rank‘𝑥) ⊆ (rank‘𝑦)))}
341, 2, 333eqtri 2825 1 Scott {𝑥𝜑} = {𝑥 ∣ (𝜑 ∧ ∀𝑦(𝜓 → (rank‘𝑥) ⊆ (rank‘𝑦)))}
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536   = wceq 1538  Ⅎwnf 1785  [wsb 2069   ∈ wcel 2111  {cab 2776  ∀wral 3106  {crab 3110   ⊆ wss 3881  ‘cfv 6325  rankcrnk 9179  Scott cscott 40986 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rab 3115  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-iota 6284  df-fv 6333  df-scott 40987 This theorem is referenced by:  scottab  40992  scottabes  40993
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