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Mirrors > Home > MPE Home > Th. List > fipwss | Structured version Visualization version GIF version |
Description: If a set is a family of subsets of some base set, then so is its finite intersection. (Contributed by Stefan O'Rear, 2-Aug-2015.) |
Ref | Expression |
---|---|
fipwss | ⊢ (𝐴 ⊆ 𝒫 𝑋 → (fi‘𝐴) ⊆ 𝒫 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fiuni 9422 | . . . . 5 ⊢ (𝐴 ∈ V → ∪ 𝐴 = ∪ (fi‘𝐴)) | |
2 | 1 | sseq1d 4013 | . . . 4 ⊢ (𝐴 ∈ V → (∪ 𝐴 ⊆ 𝑋 ↔ ∪ (fi‘𝐴) ⊆ 𝑋)) |
3 | sspwuni 5103 | . . . 4 ⊢ (𝐴 ⊆ 𝒫 𝑋 ↔ ∪ 𝐴 ⊆ 𝑋) | |
4 | sspwuni 5103 | . . . 4 ⊢ ((fi‘𝐴) ⊆ 𝒫 𝑋 ↔ ∪ (fi‘𝐴) ⊆ 𝑋) | |
5 | 2, 3, 4 | 3bitr4g 313 | . . 3 ⊢ (𝐴 ∈ V → (𝐴 ⊆ 𝒫 𝑋 ↔ (fi‘𝐴) ⊆ 𝒫 𝑋)) |
6 | 5 | biimpa 477 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐴 ⊆ 𝒫 𝑋) → (fi‘𝐴) ⊆ 𝒫 𝑋) |
7 | fvprc 6883 | . . . 4 ⊢ (¬ 𝐴 ∈ V → (fi‘𝐴) = ∅) | |
8 | 0ss 4396 | . . . 4 ⊢ ∅ ⊆ 𝒫 𝑋 | |
9 | 7, 8 | eqsstrdi 4036 | . . 3 ⊢ (¬ 𝐴 ∈ V → (fi‘𝐴) ⊆ 𝒫 𝑋) |
10 | 9 | adantr 481 | . 2 ⊢ ((¬ 𝐴 ∈ V ∧ 𝐴 ⊆ 𝒫 𝑋) → (fi‘𝐴) ⊆ 𝒫 𝑋) |
11 | 6, 10 | pm2.61ian 810 | 1 ⊢ (𝐴 ⊆ 𝒫 𝑋 → (fi‘𝐴) ⊆ 𝒫 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2106 Vcvv 3474 ⊆ wss 3948 ∅c0 4322 𝒫 cpw 4602 ∪ cuni 4908 ‘cfv 6543 ficfi 9404 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-om 7855 df-1o 8465 df-er 8702 df-en 8939 df-fin 8942 df-fi 9405 |
This theorem is referenced by: fsubbas 23370 |
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