Theorem ispisys 34153

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 6906	. . 3 ⊢ (𝑠 = 𝑆 → (fi‘𝑠) = (fi‘𝑆))
2		id 22	. . 3 ⊢ (𝑠 = 𝑆 → 𝑠 = 𝑆)
3	1, 2	sseq12d 4017	. 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 3695	1 ⊢ (𝑆 ∈ 𝑃 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (fi‘𝑆) ⊆ 𝑆))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  {crab 3436   ⊆ wss 3951  𝒫 cpw 4600  ‘cfv 6561  ficfi 9450

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 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569

This theorem is referenced by:  ispisys2  34154  sigapildsyslem  34162  sigapildsys  34163  ldgenpisyslem1  34164  ldgenpisyslem3  34166  ldgenpisys  34167