Theorem ispisys 34116

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 6920	. . 3 ⊢ (𝑠 = 𝑆 → (fi‘𝑠) = (fi‘𝑆))
2		id 22	. . 3 ⊢ (𝑠 = 𝑆 → 𝑠 = 𝑆)
3	1, 2	sseq12d 4042	. 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 3711	1 ⊢ (𝑆 ∈ 𝑃 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (fi‘𝑆) ⊆ 𝑆))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  {crab 3443   ⊆ wss 3976  𝒫 cpw 4622  ‘cfv 6573  ficfi 9479

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-iota 6525  df-fv 6581

This theorem is referenced by:  ispisys2  34117  sigapildsyslem  34125  sigapildsys  34126  ldgenpisyslem1  34127  ldgenpisyslem3  34129  ldgenpisys  34130