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Mirrors > Home > MPE Home > Th. List > fipwuni | Structured version Visualization version GIF version |
Description: The set of finite intersections of a set is contained in the powerset of the union of the elements of 𝐴. (Contributed by Mario Carneiro, 24-Nov-2013.) (Proof shortened by Mario Carneiro, 21-Mar-2015.) |
Ref | Expression |
---|---|
fipwuni | ⊢ (fi‘𝐴) ⊆ 𝒫 ∪ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uniexg 7484 | . . . . 5 ⊢ (𝐴 ∈ V → ∪ 𝐴 ∈ V) | |
2 | 1 | pwexd 5246 | . . . 4 ⊢ (𝐴 ∈ V → 𝒫 ∪ 𝐴 ∈ V) |
3 | pwuni 4835 | . . . 4 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴 | |
4 | fiss 8961 | . . . 4 ⊢ ((𝒫 ∪ 𝐴 ∈ V ∧ 𝐴 ⊆ 𝒫 ∪ 𝐴) → (fi‘𝐴) ⊆ (fi‘𝒫 ∪ 𝐴)) | |
5 | 2, 3, 4 | sylancl 589 | . . 3 ⊢ (𝐴 ∈ V → (fi‘𝐴) ⊆ (fi‘𝒫 ∪ 𝐴)) |
6 | ssinss1 4128 | . . . . . . 7 ⊢ (𝑥 ⊆ ∪ 𝐴 → (𝑥 ∩ 𝑦) ⊆ ∪ 𝐴) | |
7 | vex 3402 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
8 | 7 | elpw 4492 | . . . . . . 7 ⊢ (𝑥 ∈ 𝒫 ∪ 𝐴 ↔ 𝑥 ⊆ ∪ 𝐴) |
9 | 7 | inex1 5185 | . . . . . . . 8 ⊢ (𝑥 ∩ 𝑦) ∈ V |
10 | 9 | elpw 4492 | . . . . . . 7 ⊢ ((𝑥 ∩ 𝑦) ∈ 𝒫 ∪ 𝐴 ↔ (𝑥 ∩ 𝑦) ⊆ ∪ 𝐴) |
11 | 6, 8, 10 | 3imtr4i 295 | . . . . . 6 ⊢ (𝑥 ∈ 𝒫 ∪ 𝐴 → (𝑥 ∩ 𝑦) ∈ 𝒫 ∪ 𝐴) |
12 | 11 | adantr 484 | . . . . 5 ⊢ ((𝑥 ∈ 𝒫 ∪ 𝐴 ∧ 𝑦 ∈ 𝒫 ∪ 𝐴) → (𝑥 ∩ 𝑦) ∈ 𝒫 ∪ 𝐴) |
13 | 12 | rgen2 3115 | . . . 4 ⊢ ∀𝑥 ∈ 𝒫 ∪ 𝐴∀𝑦 ∈ 𝒫 ∪ 𝐴(𝑥 ∩ 𝑦) ∈ 𝒫 ∪ 𝐴 |
14 | inficl 8962 | . . . . 5 ⊢ (𝒫 ∪ 𝐴 ∈ V → (∀𝑥 ∈ 𝒫 ∪ 𝐴∀𝑦 ∈ 𝒫 ∪ 𝐴(𝑥 ∩ 𝑦) ∈ 𝒫 ∪ 𝐴 ↔ (fi‘𝒫 ∪ 𝐴) = 𝒫 ∪ 𝐴)) | |
15 | 2, 14 | syl 17 | . . . 4 ⊢ (𝐴 ∈ V → (∀𝑥 ∈ 𝒫 ∪ 𝐴∀𝑦 ∈ 𝒫 ∪ 𝐴(𝑥 ∩ 𝑦) ∈ 𝒫 ∪ 𝐴 ↔ (fi‘𝒫 ∪ 𝐴) = 𝒫 ∪ 𝐴)) |
16 | 13, 15 | mpbii 236 | . . 3 ⊢ (𝐴 ∈ V → (fi‘𝒫 ∪ 𝐴) = 𝒫 ∪ 𝐴) |
17 | 5, 16 | sseqtrd 3917 | . 2 ⊢ (𝐴 ∈ V → (fi‘𝐴) ⊆ 𝒫 ∪ 𝐴) |
18 | fvprc 6666 | . . 3 ⊢ (¬ 𝐴 ∈ V → (fi‘𝐴) = ∅) | |
19 | 0ss 4285 | . . 3 ⊢ ∅ ⊆ 𝒫 ∪ 𝐴 | |
20 | 18, 19 | eqsstrdi 3931 | . 2 ⊢ (¬ 𝐴 ∈ V → (fi‘𝐴) ⊆ 𝒫 ∪ 𝐴) |
21 | 17, 20 | pm2.61i 185 | 1 ⊢ (fi‘𝐴) ⊆ 𝒫 ∪ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 209 = wceq 1542 ∈ wcel 2114 ∀wral 3053 Vcvv 3398 ∩ cin 3842 ⊆ wss 3843 ∅c0 4211 𝒫 cpw 4488 ∪ cuni 4796 ‘cfv 6339 ficfi 8947 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-om 7600 df-1o 8131 df-er 8320 df-en 8556 df-fin 8559 df-fi 8948 |
This theorem is referenced by: fiuni 8965 ordtbas 21943 |
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