![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 7234 | . . . . 5 ⊢ (𝐴 ∈ V → ∪ 𝐴 ∈ V) | |
2 | 1 | pwexd 5093 | . . . 4 ⊢ (𝐴 ∈ V → 𝒫 ∪ 𝐴 ∈ V) |
3 | pwuni 4711 | . . . 4 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴 | |
4 | fiss 8620 | . . . 4 ⊢ ((𝒫 ∪ 𝐴 ∈ V ∧ 𝐴 ⊆ 𝒫 ∪ 𝐴) → (fi‘𝐴) ⊆ (fi‘𝒫 ∪ 𝐴)) | |
5 | 2, 3, 4 | sylancl 580 | . . 3 ⊢ (𝐴 ∈ V → (fi‘𝐴) ⊆ (fi‘𝒫 ∪ 𝐴)) |
6 | ssinss1 4062 | . . . . . . 7 ⊢ (𝑥 ⊆ ∪ 𝐴 → (𝑥 ∩ 𝑦) ⊆ ∪ 𝐴) | |
7 | vex 3401 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
8 | 7 | elpw 4385 | . . . . . . 7 ⊢ (𝑥 ∈ 𝒫 ∪ 𝐴 ↔ 𝑥 ⊆ ∪ 𝐴) |
9 | 7 | inex1 5038 | . . . . . . . 8 ⊢ (𝑥 ∩ 𝑦) ∈ V |
10 | 9 | elpw 4385 | . . . . . . 7 ⊢ ((𝑥 ∩ 𝑦) ∈ 𝒫 ∪ 𝐴 ↔ (𝑥 ∩ 𝑦) ⊆ ∪ 𝐴) |
11 | 6, 8, 10 | 3imtr4i 284 | . . . . . 6 ⊢ (𝑥 ∈ 𝒫 ∪ 𝐴 → (𝑥 ∩ 𝑦) ∈ 𝒫 ∪ 𝐴) |
12 | 11 | adantr 474 | . . . . 5 ⊢ ((𝑥 ∈ 𝒫 ∪ 𝐴 ∧ 𝑦 ∈ 𝒫 ∪ 𝐴) → (𝑥 ∩ 𝑦) ∈ 𝒫 ∪ 𝐴) |
13 | 12 | rgen2a 3159 | . . . 4 ⊢ ∀𝑥 ∈ 𝒫 ∪ 𝐴∀𝑦 ∈ 𝒫 ∪ 𝐴(𝑥 ∩ 𝑦) ∈ 𝒫 ∪ 𝐴 |
14 | inficl 8621 | . . . . 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 225 | . . 3 ⊢ (𝐴 ∈ V → (fi‘𝒫 ∪ 𝐴) = 𝒫 ∪ 𝐴) |
17 | 5, 16 | sseqtrd 3860 | . 2 ⊢ (𝐴 ∈ V → (fi‘𝐴) ⊆ 𝒫 ∪ 𝐴) |
18 | fvprc 6441 | . . 3 ⊢ (¬ 𝐴 ∈ V → (fi‘𝐴) = ∅) | |
19 | 0ss 4198 | . . 3 ⊢ ∅ ⊆ 𝒫 ∪ 𝐴 | |
20 | 18, 19 | syl6eqss 3874 | . 2 ⊢ (¬ 𝐴 ∈ V → (fi‘𝐴) ⊆ 𝒫 ∪ 𝐴) |
21 | 17, 20 | pm2.61i 177 | 1 ⊢ (fi‘𝐴) ⊆ 𝒫 ∪ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 198 = wceq 1601 ∈ wcel 2107 ∀wral 3090 Vcvv 3398 ∩ cin 3791 ⊆ wss 3792 ∅c0 4141 𝒫 cpw 4379 ∪ cuni 4673 ‘cfv 6137 ficfi 8606 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-oadd 7849 df-er 8028 df-en 8244 df-fin 8247 df-fi 8607 |
This theorem is referenced by: fiuni 8624 ordtbas 21415 |
Copyright terms: Public domain | W3C validator |