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Theorem scottabf 44236
Description: Value of the Scott operation at a class abstraction. Variant of scottab 44237 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 44232 . 2 Scott {𝑥𝜑} = {𝑧 ∈ {𝑥𝜑} ∣ ∀𝑤 ∈ {𝑥𝜑} (rank‘𝑧) ⊆ (rank‘𝑤)}
2 df-rab 3434 . 2 {𝑧 ∈ {𝑥𝜑} ∣ ∀𝑤 ∈ {𝑥𝜑} (rank‘𝑧) ⊆ (rank‘𝑤)} = {𝑧 ∣ (𝑧 ∈ {𝑥𝜑} ∧ ∀𝑤 ∈ {𝑥𝜑} (rank‘𝑧) ⊆ (rank‘𝑤))}
3 eqabcb 2881 . . 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 2720 . . . . . 6 𝑥 𝑧 ∈ {𝑥𝜑}
5 nfab1 2905 . . . . . . 7 𝑥{𝑥𝜑}
6 nfv 1912 . . . . . . 7 𝑥(rank‘𝑧) ⊆ (rank‘𝑤)
75, 6nfralw 3309 . . . . . 6 𝑥𝑤 ∈ {𝑥𝜑} (rank‘𝑧) ⊆ (rank‘𝑤)
84, 7nfan 1897 . . . . 5 𝑥(𝑧 ∈ {𝑥𝜑} ∧ ∀𝑤 ∈ {𝑥𝜑} (rank‘𝑧) ⊆ (rank‘𝑤))
9 vex 3482 . . . . 5 𝑧 ∈ V
10 abid 2716 . . . . . . 7 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
11 eleq1w 2822 . . . . . . 7 (𝑥 = 𝑧 → (𝑥 ∈ {𝑥𝜑} ↔ 𝑧 ∈ {𝑥𝜑}))
1210, 11bitr3id 285 . . . . . 6 (𝑥 = 𝑧 → (𝜑𝑧 ∈ {𝑥𝜑}))
13 df-clab 2713 . . . . . . . . . . . 12 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
14 scottabf.1 . . . . . . . . . . . . 13 𝑥𝜓
15 scottabf.2 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝜑𝜓))
1614, 15sbiev 2313 . . . . . . . . . . . 12 ([𝑦 / 𝑥]𝜑𝜓)
1713, 16bitr2i 276 . . . . . . . . . . 11 (𝜓𝑦 ∈ {𝑥𝜑})
18 eleq1w 2822 . . . . . . . . . . 11 (𝑦 = 𝑤 → (𝑦 ∈ {𝑥𝜑} ↔ 𝑤 ∈ {𝑥𝜑}))
1917, 18bitrid 283 . . . . . . . . . 10 (𝑦 = 𝑤 → (𝜓𝑤 ∈ {𝑥𝜑}))
2019adantl 481 . . . . . . . . 9 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜓𝑤 ∈ {𝑥𝜑}))
21 simpl 482 . . . . . . . . . . 11 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝑥 = 𝑧)
2221fveq2d 6911 . . . . . . . . . 10 ((𝑥 = 𝑧𝑦 = 𝑤) → (rank‘𝑥) = (rank‘𝑧))
23 simpr 484 . . . . . . . . . . 11 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝑦 = 𝑤)
2423fveq2d 6911 . . . . . . . . . 10 ((𝑥 = 𝑧𝑦 = 𝑤) → (rank‘𝑦) = (rank‘𝑤))
2522, 24sseq12d 4029 . . . . . . . . 9 ((𝑥 = 𝑧𝑦 = 𝑤) → ((rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝑧) ⊆ (rank‘𝑤)))
2620, 25imbi12d 344 . . . . . . . 8 ((𝑥 = 𝑧𝑦 = 𝑤) → ((𝜓 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ (𝑤 ∈ {𝑥𝜑} → (rank‘𝑧) ⊆ (rank‘𝑤))))
2726cbvaldvaw 2035 . . . . . . 7 (𝑥 = 𝑧 → (∀𝑦(𝜓 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ ∀𝑤(𝑤 ∈ {𝑥𝜑} → (rank‘𝑧) ⊆ (rank‘𝑤))))
28 df-ral 3060 . . . . . . 7 (∀𝑤 ∈ {𝑥𝜑} (rank‘𝑧) ⊆ (rank‘𝑤) ↔ ∀𝑤(𝑤 ∈ {𝑥𝜑} → (rank‘𝑧) ⊆ (rank‘𝑤)))
2927, 28bitr4di 289 . . . . . 6 (𝑥 = 𝑧 → (∀𝑦(𝜓 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ ∀𝑤 ∈ {𝑥𝜑} (rank‘𝑧) ⊆ (rank‘𝑤)))
3012, 29anbi12d 632 . . . . 5 (𝑥 = 𝑧 → ((𝜑 ∧ ∀𝑦(𝜓 → (rank‘𝑥) ⊆ (rank‘𝑦))) ↔ (𝑧 ∈ {𝑥𝜑} ∧ ∀𝑤 ∈ {𝑥𝜑} (rank‘𝑧) ⊆ (rank‘𝑤))))
318, 9, 30elabf 3676 . . . 4 (𝑧 ∈ {𝑥 ∣ (𝜑 ∧ ∀𝑦(𝜓 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ↔ (𝑧 ∈ {𝑥𝜑} ∧ ∀𝑤 ∈ {𝑥𝜑} (rank‘𝑧) ⊆ (rank‘𝑤)))
3231bicomi 224 . . 3 ((𝑧 ∈ {𝑥𝜑} ∧ ∀𝑤 ∈ {𝑥𝜑} (rank‘𝑧) ⊆ (rank‘𝑤)) ↔ 𝑧 ∈ {𝑥 ∣ (𝜑 ∧ ∀𝑦(𝜓 → (rank‘𝑥) ⊆ (rank‘𝑦)))})
333, 32mpgbir 1796 . 2 {𝑧 ∣ (𝑧 ∈ {𝑥𝜑} ∧ ∀𝑤 ∈ {𝑥𝜑} (rank‘𝑧) ⊆ (rank‘𝑤))} = {𝑥 ∣ (𝜑 ∧ ∀𝑦(𝜓 → (rank‘𝑥) ⊆ (rank‘𝑦)))}
341, 2, 333eqtri 2767 1 Scott {𝑥𝜑} = {𝑥 ∣ (𝜑 ∧ ∀𝑦(𝜓 → (rank‘𝑥) ⊆ (rank‘𝑦)))}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1535   = wceq 1537  wnf 1780  [wsb 2062  wcel 2106  {cab 2712  wral 3059  {crab 3433  wss 3963  cfv 6563  rankcrnk 9801  Scott cscott 44231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-scott 44232
This theorem is referenced by:  scottab  44237  scottabes  44238
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