Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
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 6774 | . . 3 ⊢ (𝑠 = 𝑆 → (fi‘𝑠) = (fi‘𝑆)) | |
2 | id 22 | . . 3 ⊢ (𝑠 = 𝑆 → 𝑠 = 𝑆) | |
3 | 1, 2 | sseq12d 3954 | . 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 3627 | 1 ⊢ (𝑆 ∈ 𝑃 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (fi‘𝑆) ⊆ 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 {crab 3068 ⊆ wss 3887 𝒫 cpw 4533 ‘cfv 6433 ficfi 9169 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-iota 6391 df-fv 6441 |
This theorem is referenced by: ispisys2 32121 sigapildsyslem 32129 sigapildsys 32130 ldgenpisyslem1 32131 ldgenpisyslem3 32133 ldgenpisys 32134 |
Copyright terms: Public domain | W3C validator |