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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 7761 | . . . . 5 ⊢ (𝐴 ∈ V → ∪ 𝐴 ∈ V) | |
| 2 | 1 | pwexd 5378 | . . . 4 ⊢ (𝐴 ∈ V → 𝒫 ∪ 𝐴 ∈ V) | 
| 3 | pwuni 4944 | . . . 4 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴 | |
| 4 | fiss 9465 | . . . 4 ⊢ ((𝒫 ∪ 𝐴 ∈ V ∧ 𝐴 ⊆ 𝒫 ∪ 𝐴) → (fi‘𝐴) ⊆ (fi‘𝒫 ∪ 𝐴)) | |
| 5 | 2, 3, 4 | sylancl 586 | . . 3 ⊢ (𝐴 ∈ V → (fi‘𝐴) ⊆ (fi‘𝒫 ∪ 𝐴)) | 
| 6 | ssinss1 4245 | . . . . . . 7 ⊢ (𝑥 ⊆ ∪ 𝐴 → (𝑥 ∩ 𝑦) ⊆ ∪ 𝐴) | |
| 7 | vex 3483 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
| 8 | 7 | elpw 4603 | . . . . . . 7 ⊢ (𝑥 ∈ 𝒫 ∪ 𝐴 ↔ 𝑥 ⊆ ∪ 𝐴) | 
| 9 | 7 | inex1 5316 | . . . . . . . 8 ⊢ (𝑥 ∩ 𝑦) ∈ V | 
| 10 | 9 | elpw 4603 | . . . . . . 7 ⊢ ((𝑥 ∩ 𝑦) ∈ 𝒫 ∪ 𝐴 ↔ (𝑥 ∩ 𝑦) ⊆ ∪ 𝐴) | 
| 11 | 6, 8, 10 | 3imtr4i 292 | . . . . . 6 ⊢ (𝑥 ∈ 𝒫 ∪ 𝐴 → (𝑥 ∩ 𝑦) ∈ 𝒫 ∪ 𝐴) | 
| 12 | 11 | adantr 480 | . . . . 5 ⊢ ((𝑥 ∈ 𝒫 ∪ 𝐴 ∧ 𝑦 ∈ 𝒫 ∪ 𝐴) → (𝑥 ∩ 𝑦) ∈ 𝒫 ∪ 𝐴) | 
| 13 | 12 | rgen2 3198 | . . . 4 ⊢ ∀𝑥 ∈ 𝒫 ∪ 𝐴∀𝑦 ∈ 𝒫 ∪ 𝐴(𝑥 ∩ 𝑦) ∈ 𝒫 ∪ 𝐴 | 
| 14 | inficl 9466 | . . . . 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 233 | . . 3 ⊢ (𝐴 ∈ V → (fi‘𝒫 ∪ 𝐴) = 𝒫 ∪ 𝐴) | 
| 17 | 5, 16 | sseqtrd 4019 | . 2 ⊢ (𝐴 ∈ V → (fi‘𝐴) ⊆ 𝒫 ∪ 𝐴) | 
| 18 | fvprc 6897 | . . 3 ⊢ (¬ 𝐴 ∈ V → (fi‘𝐴) = ∅) | |
| 19 | 0ss 4399 | . . 3 ⊢ ∅ ⊆ 𝒫 ∪ 𝐴 | |
| 20 | 18, 19 | eqsstrdi 4027 | . 2 ⊢ (¬ 𝐴 ∈ V → (fi‘𝐴) ⊆ 𝒫 ∪ 𝐴) | 
| 21 | 17, 20 | pm2.61i 182 | 1 ⊢ (fi‘𝐴) ⊆ 𝒫 ∪ 𝐴 | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 ↔ wb 206 = wceq 1539 ∈ wcel 2107 ∀wral 3060 Vcvv 3479 ∩ cin 3949 ⊆ wss 3950 ∅c0 4332 𝒫 cpw 4599 ∪ cuni 4906 ‘cfv 6560 ficfi 9451 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-om 7889 df-1o 8507 df-2o 8508 df-en 8987 df-fin 8990 df-fi 9452 | 
| This theorem is referenced by: fiuni 9469 ordtbas 23201 | 
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