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Mirrors > Home > MPE Home > Th. List > fival | Structured version Visualization version GIF version |
Description: The set of all the finite intersections of the elements of 𝐴. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 24-Nov-2013.) |
Ref | Expression |
---|---|
fival | ⊢ (𝐴 ∈ 𝑉 → (fi‘𝐴) = {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = ∩ 𝑥}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-fi 9449 | . 2 ⊢ fi = (𝑧 ∈ V ↦ {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝑧 ∩ Fin)𝑦 = ∩ 𝑥}) | |
2 | pweq 4619 | . . . . 5 ⊢ (𝑧 = 𝐴 → 𝒫 𝑧 = 𝒫 𝐴) | |
3 | 2 | ineq1d 4227 | . . . 4 ⊢ (𝑧 = 𝐴 → (𝒫 𝑧 ∩ Fin) = (𝒫 𝐴 ∩ Fin)) |
4 | 3 | rexeqdv 3325 | . . 3 ⊢ (𝑧 = 𝐴 → (∃𝑥 ∈ (𝒫 𝑧 ∩ Fin)𝑦 = ∩ 𝑥 ↔ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = ∩ 𝑥)) |
5 | 4 | abbidv 2806 | . 2 ⊢ (𝑧 = 𝐴 → {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝑧 ∩ Fin)𝑦 = ∩ 𝑥} = {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = ∩ 𝑥}) |
6 | elex 3499 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
7 | simpr 484 | . . . . . . 7 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 = ∩ 𝑥) → 𝑦 = ∩ 𝑥) | |
8 | elinel1 4211 | . . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → 𝑥 ∈ 𝒫 𝐴) | |
9 | 8 | elpwid 4614 | . . . . . . . 8 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → 𝑥 ⊆ 𝐴) |
10 | eqvisset 3498 | . . . . . . . . 9 ⊢ (𝑦 = ∩ 𝑥 → ∩ 𝑥 ∈ V) | |
11 | intex 5350 | . . . . . . . . 9 ⊢ (𝑥 ≠ ∅ ↔ ∩ 𝑥 ∈ V) | |
12 | 10, 11 | sylibr 234 | . . . . . . . 8 ⊢ (𝑦 = ∩ 𝑥 → 𝑥 ≠ ∅) |
13 | intssuni2 4978 | . . . . . . . 8 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∩ 𝑥 ⊆ ∪ 𝐴) | |
14 | 9, 12, 13 | syl2an 596 | . . . . . . 7 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 = ∩ 𝑥) → ∩ 𝑥 ⊆ ∪ 𝐴) |
15 | 7, 14 | eqsstrd 4034 | . . . . . 6 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 = ∩ 𝑥) → 𝑦 ⊆ ∪ 𝐴) |
16 | velpw 4610 | . . . . . 6 ⊢ (𝑦 ∈ 𝒫 ∪ 𝐴 ↔ 𝑦 ⊆ ∪ 𝐴) | |
17 | 15, 16 | sylibr 234 | . . . . 5 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 = ∩ 𝑥) → 𝑦 ∈ 𝒫 ∪ 𝐴) |
18 | 17 | rexlimiva 3145 | . . . 4 ⊢ (∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = ∩ 𝑥 → 𝑦 ∈ 𝒫 ∪ 𝐴) |
19 | 18 | abssi 4080 | . . 3 ⊢ {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = ∩ 𝑥} ⊆ 𝒫 ∪ 𝐴 |
20 | uniexg 7759 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V) | |
21 | 20 | pwexd 5385 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝒫 ∪ 𝐴 ∈ V) |
22 | ssexg 5329 | . . 3 ⊢ (({𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = ∩ 𝑥} ⊆ 𝒫 ∪ 𝐴 ∧ 𝒫 ∪ 𝐴 ∈ V) → {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = ∩ 𝑥} ∈ V) | |
23 | 19, 21, 22 | sylancr 587 | . 2 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = ∩ 𝑥} ∈ V) |
24 | 1, 5, 6, 23 | fvmptd3 7039 | 1 ⊢ (𝐴 ∈ 𝑉 → (fi‘𝐴) = {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = ∩ 𝑥}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 {cab 2712 ≠ wne 2938 ∃wrex 3068 Vcvv 3478 ∩ cin 3962 ⊆ wss 3963 ∅c0 4339 𝒫 cpw 4605 ∪ cuni 4912 ∩ cint 4951 ‘cfv 6563 Fincfn 8984 ficfi 9448 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-fi 9449 |
This theorem is referenced by: elfi 9451 fi0 9458 |
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