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Theorem ispisys 29338
Description: The property of being a pi-system. (Contributed by Thierry Arnoux, 10-Jun-2020.)
Hypothesis
Ref Expression
ispisys.p 𝑃 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (fi‘𝑠) ⊆ 𝑠}
Assertion
Ref Expression
ispisys (𝑆𝑃 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (fi‘𝑆) ⊆ 𝑆))
Distinct variable groups:   𝑂,𝑠   𝑆,𝑠
Allowed substitution hint:   𝑃(𝑠)

Proof of Theorem ispisys
StepHypRef Expression
1 fveq2 5987 . . 3 (𝑠 = 𝑆 → (fi‘𝑠) = (fi‘𝑆))
2 id 22 . . 3 (𝑠 = 𝑆𝑠 = 𝑆)
31, 2sseq12d 3501 . 2 (𝑠 = 𝑆 → ((fi‘𝑠) ⊆ 𝑠 ↔ (fi‘𝑆) ⊆ 𝑆))
4 ispisys.p . 2 𝑃 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (fi‘𝑠) ⊆ 𝑠}
53, 4elrab2 3237 1 (𝑆𝑃 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (fi‘𝑆) ⊆ 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wb 194  wa 382   = wceq 1474  wcel 1938  {crab 2804  wss 3444  𝒫 cpw 4011  cfv 5689  ficfi 8075
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-rex 2806  df-rab 2809  df-v 3079  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-sn 4029  df-pr 4031  df-op 4035  df-uni 4271  df-br 4482  df-iota 5653  df-fv 5697
This theorem is referenced by:  ispisys2  29339  sigapildsyslem  29347  sigapildsys  29348  ldgenpisyslem1  29349  ldgenpisyslem3  29351  ldgenpisys  29352
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