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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 6906 | . . 3 ⊢ (𝑠 = 𝑆 → (fi‘𝑠) = (fi‘𝑆)) | |
| 2 | id 22 | . . 3 ⊢ (𝑠 = 𝑆 → 𝑠 = 𝑆) | |
| 3 | 1, 2 | sseq12d 4017 | . 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 3695 | 1 ⊢ (𝑆 ∈ 𝑃 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (fi‘𝑆) ⊆ 𝑆)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {crab 3436 ⊆ wss 3951 𝒫 cpw 4600 ‘cfv 6561 ficfi 9450 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 |
| This theorem is referenced by: ispisys2 34154 sigapildsyslem 34162 sigapildsys 34163 ldgenpisyslem1 34164 ldgenpisyslem3 34166 ldgenpisys 34167 |
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