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Theorem scottexs 9928
Description: Theorem scheme version of scottex 9926. The collection of all 𝑥 of minimum rank such that 𝜑(𝑥) is true, is a set. (Contributed by NM, 13-Oct-2003.)
Assertion
Ref Expression
scottexs {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ∈ V
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem scottexs
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 nfcv 2904 . . . 4 𝑧{𝑥𝜑}
2 nfab1 2906 . . . 4 𝑥{𝑥𝜑}
3 nfv 1913 . . . . 5 𝑥(rank‘𝑧) ⊆ (rank‘𝑦)
42, 3nfralw 3310 . . . 4 𝑥𝑦 ∈ {𝑥𝜑} (rank‘𝑧) ⊆ (rank‘𝑦)
5 nfv 1913 . . . 4 𝑧𝑦 ∈ {𝑥𝜑} (rank‘𝑥) ⊆ (rank‘𝑦)
6 fveq2 6905 . . . . . 6 (𝑧 = 𝑥 → (rank‘𝑧) = (rank‘𝑥))
76sseq1d 4014 . . . . 5 (𝑧 = 𝑥 → ((rank‘𝑧) ⊆ (rank‘𝑦) ↔ (rank‘𝑥) ⊆ (rank‘𝑦)))
87ralbidv 3177 . . . 4 (𝑧 = 𝑥 → (∀𝑦 ∈ {𝑥𝜑} (rank‘𝑧) ⊆ (rank‘𝑦) ↔ ∀𝑦 ∈ {𝑥𝜑} (rank‘𝑥) ⊆ (rank‘𝑦)))
91, 2, 4, 5, 8cbvrabw 3472 . . 3 {𝑧 ∈ {𝑥𝜑} ∣ ∀𝑦 ∈ {𝑥𝜑} (rank‘𝑧) ⊆ (rank‘𝑦)} = {𝑥 ∈ {𝑥𝜑} ∣ ∀𝑦 ∈ {𝑥𝜑} (rank‘𝑥) ⊆ (rank‘𝑦)}
10 df-rab 3436 . . 3 {𝑥 ∈ {𝑥𝜑} ∣ ∀𝑦 ∈ {𝑥𝜑} (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑥 ∣ (𝑥 ∈ {𝑥𝜑} ∧ ∀𝑦 ∈ {𝑥𝜑} (rank‘𝑥) ⊆ (rank‘𝑦))}
11 abid 2717 . . . . 5 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
12 df-ral 3061 . . . . . 6 (∀𝑦 ∈ {𝑥𝜑} (rank‘𝑥) ⊆ (rank‘𝑦) ↔ ∀𝑦(𝑦 ∈ {𝑥𝜑} → (rank‘𝑥) ⊆ (rank‘𝑦)))
13 df-sbc 3788 . . . . . . . 8 ([𝑦 / 𝑥]𝜑𝑦 ∈ {𝑥𝜑})
1413imbi1i 349 . . . . . . 7 (([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ (𝑦 ∈ {𝑥𝜑} → (rank‘𝑥) ⊆ (rank‘𝑦)))
1514albii 1818 . . . . . 6 (∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ ∀𝑦(𝑦 ∈ {𝑥𝜑} → (rank‘𝑥) ⊆ (rank‘𝑦)))
1612, 15bitr4i 278 . . . . 5 (∀𝑦 ∈ {𝑥𝜑} (rank‘𝑥) ⊆ (rank‘𝑦) ↔ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))
1711, 16anbi12i 628 . . . 4 ((𝑥 ∈ {𝑥𝜑} ∧ ∀𝑦 ∈ {𝑥𝜑} (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦))))
1817abbii 2808 . . 3 {𝑥 ∣ (𝑥 ∈ {𝑥𝜑} ∧ ∀𝑦 ∈ {𝑥𝜑} (rank‘𝑥) ⊆ (rank‘𝑦))} = {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))}
199, 10, 183eqtri 2768 . 2 {𝑧 ∈ {𝑥𝜑} ∣ ∀𝑦 ∈ {𝑥𝜑} (rank‘𝑧) ⊆ (rank‘𝑦)} = {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))}
20 scottex 9926 . 2 {𝑧 ∈ {𝑥𝜑} ∣ ∀𝑦 ∈ {𝑥𝜑} (rank‘𝑧) ⊆ (rank‘𝑦)} ∈ V
2119, 20eqeltrri 2837 1 {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ∈ V
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1537  wcel 2107  {cab 2713  wral 3060  {crab 3435  Vcvv 3479  [wsbc 3787  wss 3950  cfv 6560  rankcrnk 9804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-reg 9633  ax-inf2 9682
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-om 7889  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-r1 9805  df-rank 9806
This theorem is referenced by:  hta  9938
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