Theorem ispisys 34133

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 6907	. . 3 ⊢ (𝑠 = 𝑆 → (fi‘𝑠) = (fi‘𝑆))
2		id 22	. . 3 ⊢ (𝑠 = 𝑆 → 𝑠 = 𝑆)
3	1, 2	sseq12d 4029	. 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 3698	1 ⊢ (𝑆 ∈ 𝑃 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (fi‘𝑆) ⊆ 𝑆))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106  {crab 3433   ⊆ wss 3963  𝒫 cpw 4605  ‘cfv 6563  ficfi 9448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571

This theorem is referenced by:  ispisys2  34134  sigapildsyslem  34142  sigapildsys  34143  ldgenpisyslem1  34144  ldgenpisyslem3  34146  ldgenpisys  34147