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Theorem ispisys 30555
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 6332 . . 3 (𝑠 = 𝑆 → (fi‘𝑠) = (fi‘𝑆))
2 id 22 . . 3 (𝑠 = 𝑆𝑠 = 𝑆)
31, 2sseq12d 3783 . 2 (𝑠 = 𝑆 → ((fi‘𝑠) ⊆ 𝑠 ↔ (fi‘𝑆) ⊆ 𝑆))
4 ispisys.p . 2 𝑃 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (fi‘𝑠) ⊆ 𝑠}
53, 4elrab2 3518 1 (𝑆𝑃 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (fi‘𝑆) ⊆ 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382   = wceq 1631  wcel 2145  {crab 3065  wss 3723  𝒫 cpw 4297  cfv 6031  ficfi 8472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-iota 5994  df-fv 6039
This theorem is referenced by:  ispisys2  30556  sigapildsyslem  30564  sigapildsys  30565  ldgenpisyslem1  30566  ldgenpisyslem3  30568  ldgenpisys  30569
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