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Mirrors > Home > MPE Home > Th. List > Mathboxes > ispisys | Structured version Visualization version GIF version |
Description: The property of being a pi-system. (Contributed by Thierry Arnoux, 10-Jun-2020.) |
Ref | Expression |
---|---|
ispisys.p | ⊢ 𝑃 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (fi‘𝑠) ⊆ 𝑠} |
Ref | Expression |
---|---|
ispisys | ⊢ (𝑆 ∈ 𝑃 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (fi‘𝑆) ⊆ 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6677 | . . 3 ⊢ (𝑠 = 𝑆 → (fi‘𝑠) = (fi‘𝑆)) | |
2 | id 22 | . . 3 ⊢ (𝑠 = 𝑆 → 𝑠 = 𝑆) | |
3 | 1, 2 | sseq12d 3911 | . 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 3592 | 1 ⊢ (𝑆 ∈ 𝑃 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (fi‘𝑆) ⊆ 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2114 {crab 3058 ⊆ wss 3844 𝒫 cpw 4489 ‘cfv 6340 ficfi 8950 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2711 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-ex 1787 df-sb 2075 df-clab 2718 df-cleq 2731 df-clel 2812 df-rab 3063 df-v 3401 df-un 3849 df-in 3851 df-ss 3861 df-sn 4518 df-pr 4520 df-op 4524 df-uni 4798 df-br 5032 df-iota 6298 df-fv 6348 |
This theorem is referenced by: ispisys2 31694 sigapildsyslem 31702 sigapildsys 31703 ldgenpisyslem1 31704 ldgenpisyslem3 31706 ldgenpisys 31707 |
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