Theorem ispisys 34296

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 6840	. . 3 ⊢ (𝑠 = 𝑆 → (fi‘𝑠) = (fi‘𝑆))
2		id 22	. . 3 ⊢ (𝑠 = 𝑆 → 𝑠 = 𝑆)
3	1, 2	sseq12d 3955	. 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 3637	1 ⊢ (𝑆 ∈ 𝑃 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (fi‘𝑆) ⊆ 𝑆))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {crab 3389   ⊆ wss 3889  𝒫 cpw 4541  ‘cfv 6498  ficfi 9323

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-iota 6454  df-fv 6506

This theorem is referenced by:  ispisys2  34297  sigapildsyslem  34305  sigapildsys  34306  ldgenpisyslem1  34307  ldgenpisyslem3  34309  ldgenpisys  34310