Theorem ispisys 34148

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	⊢ (𝑆 ∈ 𝑃 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (fi‘𝑆) ⊆ 𝑆))

Distinct variable groups:   𝑂,𝑠   𝑆,𝑠

Allowed substitution hint:   𝑃(𝑠)

Proof of Theorem ispisys

Step	Hyp	Ref	Expression
1		fveq2 6860	. . 3 ⊢ (𝑠 = 𝑆 → (fi‘𝑠) = (fi‘𝑆))
2		id 22	. . 3 ⊢ (𝑠 = 𝑆 → 𝑠 = 𝑆)
3	1, 2	sseq12d 3982	. 2 ⊢ (𝑠 = 𝑆 → ((fi‘𝑠) ⊆ 𝑠 ↔ (fi‘𝑆) ⊆ 𝑆))
4		ispisys.p	. 2 ⊢ 𝑃 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (fi‘𝑠) ⊆ 𝑠}
5	3, 4	elrab2 3664	1 ⊢ (𝑆 ∈ 𝑃 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (fi‘𝑆) ⊆ 𝑆))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {crab 3408   ⊆ wss 3916  𝒫 cpw 4565  ‘cfv 6513  ficfi 9367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5110  df-iota 6466  df-fv 6521

This theorem is referenced by:  ispisys2  34149  sigapildsyslem  34157  sigapildsys  34158  ldgenpisyslem1  34159  ldgenpisyslem3  34161  ldgenpisys  34162