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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 9372 | . . . . 5 ⊢ (𝐴 ∈ V → ∪ 𝐴 = ∪ (fi‘𝐴)) | |
2 | 1 | sseq1d 3979 | . . . 4 ⊢ (𝐴 ∈ V → (∪ 𝐴 ⊆ 𝑋 ↔ ∪ (fi‘𝐴) ⊆ 𝑋)) |
3 | sspwuni 5064 | . . . 4 ⊢ (𝐴 ⊆ 𝒫 𝑋 ↔ ∪ 𝐴 ⊆ 𝑋) | |
4 | sspwuni 5064 | . . . 4 ⊢ ((fi‘𝐴) ⊆ 𝒫 𝑋 ↔ ∪ (fi‘𝐴) ⊆ 𝑋) | |
5 | 2, 3, 4 | 3bitr4g 314 | . . 3 ⊢ (𝐴 ∈ V → (𝐴 ⊆ 𝒫 𝑋 ↔ (fi‘𝐴) ⊆ 𝒫 𝑋)) |
6 | 5 | biimpa 478 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐴 ⊆ 𝒫 𝑋) → (fi‘𝐴) ⊆ 𝒫 𝑋) |
7 | fvprc 6838 | . . . 4 ⊢ (¬ 𝐴 ∈ V → (fi‘𝐴) = ∅) | |
8 | 0ss 4360 | . . . 4 ⊢ ∅ ⊆ 𝒫 𝑋 | |
9 | 7, 8 | eqsstrdi 4002 | . . 3 ⊢ (¬ 𝐴 ∈ V → (fi‘𝐴) ⊆ 𝒫 𝑋) |
10 | 9 | adantr 482 | . 2 ⊢ ((¬ 𝐴 ∈ V ∧ 𝐴 ⊆ 𝒫 𝑋) → (fi‘𝐴) ⊆ 𝒫 𝑋) |
11 | 6, 10 | pm2.61ian 811 | 1 ⊢ (𝐴 ⊆ 𝒫 𝑋 → (fi‘𝐴) ⊆ 𝒫 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2107 Vcvv 3447 ⊆ wss 3914 ∅c0 4286 𝒫 cpw 4564 ∪ cuni 4869 ‘cfv 6500 ficfi 9354 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-om 7807 df-1o 8416 df-er 8654 df-en 8890 df-fin 8893 df-fi 9355 |
This theorem is referenced by: fsubbas 23241 |
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