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Theorem ispisys 31693
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 6677 . . 3 (𝑠 = 𝑆 → (fi‘𝑠) = (fi‘𝑆))
2 id 22 . . 3 (𝑠 = 𝑆𝑠 = 𝑆)
31, 2sseq12d 3911 . 2 (𝑠 = 𝑆 → ((fi‘𝑠) ⊆ 𝑠 ↔ (fi‘𝑆) ⊆ 𝑆))
4 ispisys.p . 2 𝑃 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (fi‘𝑠) ⊆ 𝑠}
53, 4elrab2 3592 1 (𝑆𝑃 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (fi‘𝑆) ⊆ 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1542  wcel 2114  {crab 3058  wss 3844  𝒫 cpw 4489  cfv 6340  ficfi 8950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2711
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-rab 3063  df-v 3401  df-un 3849  df-in 3851  df-ss 3861  df-sn 4518  df-pr 4520  df-op 4524  df-uni 4798  df-br 5032  df-iota 6298  df-fv 6348
This theorem is referenced by:  ispisys2  31694  sigapildsyslem  31702  sigapildsys  31703  ldgenpisyslem1  31704  ldgenpisyslem3  31706  ldgenpisys  31707
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