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Theorem ispisys 30189
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 6178 . . 3 (𝑠 = 𝑆 → (fi‘𝑠) = (fi‘𝑆))
2 id 22 . . 3 (𝑠 = 𝑆𝑠 = 𝑆)
31, 2sseq12d 3626 . 2 (𝑠 = 𝑆 → ((fi‘𝑠) ⊆ 𝑠 ↔ (fi‘𝑆) ⊆ 𝑆))
4 ispisys.p . 2 𝑃 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (fi‘𝑠) ⊆ 𝑠}
53, 4elrab2 3360 1 (𝑆𝑃 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (fi‘𝑆) ⊆ 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1481  wcel 1988  {crab 2913  wss 3567  𝒫 cpw 4149  cfv 5876  ficfi 8301
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-iota 5839  df-fv 5884
This theorem is referenced by:  ispisys2  30190  sigapildsyslem  30198  sigapildsys  30199  ldgenpisyslem1  30200  ldgenpisyslem3  30202  ldgenpisys  30203
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