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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 6672 | . . 3 ⊢ (𝑠 = 𝑆 → (fi‘𝑠) = (fi‘𝑆)) | |
2 | id 22 | . . 3 ⊢ (𝑠 = 𝑆 → 𝑠 = 𝑆) | |
3 | 1, 2 | sseq12d 4002 | . 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 3685 | 1 ⊢ (𝑆 ∈ 𝑃 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (fi‘𝑆) ⊆ 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 {crab 3144 ⊆ wss 3938 𝒫 cpw 4541 ‘cfv 6357 ficfi 8876 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-iota 6316 df-fv 6365 |
This theorem is referenced by: ispisys2 31414 sigapildsyslem 31422 sigapildsys 31423 ldgenpisyslem1 31424 ldgenpisyslem3 31426 ldgenpisys 31427 |
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