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Theorem scottexs 9925
Description: Theorem scheme version of scottex 9923. 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 2903 . . . 4 𝑧{𝑥𝜑}
2 nfab1 2905 . . . 4 𝑥{𝑥𝜑}
3 nfv 1912 . . . . 5 𝑥(rank‘𝑧) ⊆ (rank‘𝑦)
42, 3nfralw 3309 . . . 4 𝑥𝑦 ∈ {𝑥𝜑} (rank‘𝑧) ⊆ (rank‘𝑦)
5 nfv 1912 . . . 4 𝑧𝑦 ∈ {𝑥𝜑} (rank‘𝑥) ⊆ (rank‘𝑦)
6 fveq2 6907 . . . . . 6 (𝑧 = 𝑥 → (rank‘𝑧) = (rank‘𝑥))
76sseq1d 4027 . . . . 5 (𝑧 = 𝑥 → ((rank‘𝑧) ⊆ (rank‘𝑦) ↔ (rank‘𝑥) ⊆ (rank‘𝑦)))
87ralbidv 3176 . . . 4 (𝑧 = 𝑥 → (∀𝑦 ∈ {𝑥𝜑} (rank‘𝑧) ⊆ (rank‘𝑦) ↔ ∀𝑦 ∈ {𝑥𝜑} (rank‘𝑥) ⊆ (rank‘𝑦)))
91, 2, 4, 5, 8cbvrabw 3471 . . 3 {𝑧 ∈ {𝑥𝜑} ∣ ∀𝑦 ∈ {𝑥𝜑} (rank‘𝑧) ⊆ (rank‘𝑦)} = {𝑥 ∈ {𝑥𝜑} ∣ ∀𝑦 ∈ {𝑥𝜑} (rank‘𝑥) ⊆ (rank‘𝑦)}
10 df-rab 3434 . . 3 {𝑥 ∈ {𝑥𝜑} ∣ ∀𝑦 ∈ {𝑥𝜑} (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑥 ∣ (𝑥 ∈ {𝑥𝜑} ∧ ∀𝑦 ∈ {𝑥𝜑} (rank‘𝑥) ⊆ (rank‘𝑦))}
11 abid 2716 . . . . 5 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
12 df-ral 3060 . . . . . 6 (∀𝑦 ∈ {𝑥𝜑} (rank‘𝑥) ⊆ (rank‘𝑦) ↔ ∀𝑦(𝑦 ∈ {𝑥𝜑} → (rank‘𝑥) ⊆ (rank‘𝑦)))
13 df-sbc 3792 . . . . . . . 8 ([𝑦 / 𝑥]𝜑𝑦 ∈ {𝑥𝜑})
1413imbi1i 349 . . . . . . 7 (([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ (𝑦 ∈ {𝑥𝜑} → (rank‘𝑥) ⊆ (rank‘𝑦)))
1514albii 1816 . . . . . 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 2807 . . 3 {𝑥 ∣ (𝑥 ∈ {𝑥𝜑} ∧ ∀𝑦 ∈ {𝑥𝜑} (rank‘𝑥) ⊆ (rank‘𝑦))} = {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))}
199, 10, 183eqtri 2767 . 2 {𝑧 ∈ {𝑥𝜑} ∣ ∀𝑦 ∈ {𝑥𝜑} (rank‘𝑧) ⊆ (rank‘𝑦)} = {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))}
20 scottex 9923 . 2 {𝑧 ∈ {𝑥𝜑} ∣ ∀𝑦 ∈ {𝑥𝜑} (rank‘𝑧) ⊆ (rank‘𝑦)} ∈ V
2119, 20eqeltrri 2836 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 1535  wcel 2106  {cab 2712  wral 3059  {crab 3433  Vcvv 3478  [wsbc 3791  wss 3963  cfv 6563  rankcrnk 9801
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  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-reg 9630  ax-inf2 9679
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-r1 9802  df-rank 9803
This theorem is referenced by:  hta  9935
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