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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 8217 | . . . . 5 ⊢ (𝐴 ∈ V → ∪ 𝐴 = ∪ (fi‘𝐴)) | |
2 | 1 | sseq1d 3595 | . . . 4 ⊢ (𝐴 ∈ V → (∪ 𝐴 ⊆ 𝑋 ↔ ∪ (fi‘𝐴) ⊆ 𝑋)) |
3 | sspwuni 4547 | . . . 4 ⊢ (𝐴 ⊆ 𝒫 𝑋 ↔ ∪ 𝐴 ⊆ 𝑋) | |
4 | sspwuni 4547 | . . . 4 ⊢ ((fi‘𝐴) ⊆ 𝒫 𝑋 ↔ ∪ (fi‘𝐴) ⊆ 𝑋) | |
5 | 2, 3, 4 | 3bitr4g 302 | . . 3 ⊢ (𝐴 ∈ V → (𝐴 ⊆ 𝒫 𝑋 ↔ (fi‘𝐴) ⊆ 𝒫 𝑋)) |
6 | 5 | biimpa 500 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐴 ⊆ 𝒫 𝑋) → (fi‘𝐴) ⊆ 𝒫 𝑋) |
7 | fvprc 6097 | . . . 4 ⊢ (¬ 𝐴 ∈ V → (fi‘𝐴) = ∅) | |
8 | 0ss 3924 | . . . 4 ⊢ ∅ ⊆ 𝒫 𝑋 | |
9 | 7, 8 | syl6eqss 3618 | . . 3 ⊢ (¬ 𝐴 ∈ V → (fi‘𝐴) ⊆ 𝒫 𝑋) |
10 | 9 | adantr 480 | . 2 ⊢ ((¬ 𝐴 ∈ V ∧ 𝐴 ⊆ 𝒫 𝑋) → (fi‘𝐴) ⊆ 𝒫 𝑋) |
11 | 6, 10 | pm2.61ian 827 | 1 ⊢ (𝐴 ⊆ 𝒫 𝑋 → (fi‘𝐴) ⊆ 𝒫 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 1977 Vcvv 3173 ⊆ wss 3540 ∅c0 3874 𝒫 cpw 4108 ∪ cuni 4372 ‘cfv 5804 ficfi 8199 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-fin 7845 df-fi 8200 |
This theorem is referenced by: fsubbas 21481 |
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