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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 8863 | . 2 ⊢ fi = (𝑧 ∈ V ↦ {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝑧 ∩ Fin)𝑦 = ∩ 𝑥}) | |
2 | pweq 4538 | . . . . 5 ⊢ (𝑧 = 𝐴 → 𝒫 𝑧 = 𝒫 𝐴) | |
3 | 2 | ineq1d 4185 | . . . 4 ⊢ (𝑧 = 𝐴 → (𝒫 𝑧 ∩ Fin) = (𝒫 𝐴 ∩ Fin)) |
4 | 3 | rexeqdv 3414 | . . 3 ⊢ (𝑧 = 𝐴 → (∃𝑥 ∈ (𝒫 𝑧 ∩ Fin)𝑦 = ∩ 𝑥 ↔ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = ∩ 𝑥)) |
5 | 4 | abbidv 2882 | . 2 ⊢ (𝑧 = 𝐴 → {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝑧 ∩ Fin)𝑦 = ∩ 𝑥} = {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = ∩ 𝑥}) |
6 | elex 3510 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
7 | simpr 485 | . . . . . . 7 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 = ∩ 𝑥) → 𝑦 = ∩ 𝑥) | |
8 | elinel1 4169 | . . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → 𝑥 ∈ 𝒫 𝐴) | |
9 | 8 | elpwid 4549 | . . . . . . . 8 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → 𝑥 ⊆ 𝐴) |
10 | eqvisset 3509 | . . . . . . . . 9 ⊢ (𝑦 = ∩ 𝑥 → ∩ 𝑥 ∈ V) | |
11 | intex 5231 | . . . . . . . . 9 ⊢ (𝑥 ≠ ∅ ↔ ∩ 𝑥 ∈ V) | |
12 | 10, 11 | sylibr 235 | . . . . . . . 8 ⊢ (𝑦 = ∩ 𝑥 → 𝑥 ≠ ∅) |
13 | intssuni2 4892 | . . . . . . . 8 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∩ 𝑥 ⊆ ∪ 𝐴) | |
14 | 9, 12, 13 | syl2an 595 | . . . . . . 7 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 = ∩ 𝑥) → ∩ 𝑥 ⊆ ∪ 𝐴) |
15 | 7, 14 | eqsstrd 4002 | . . . . . 6 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 = ∩ 𝑥) → 𝑦 ⊆ ∪ 𝐴) |
16 | velpw 4543 | . . . . . 6 ⊢ (𝑦 ∈ 𝒫 ∪ 𝐴 ↔ 𝑦 ⊆ ∪ 𝐴) | |
17 | 15, 16 | sylibr 235 | . . . . 5 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 = ∩ 𝑥) → 𝑦 ∈ 𝒫 ∪ 𝐴) |
18 | 17 | rexlimiva 3278 | . . . 4 ⊢ (∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = ∩ 𝑥 → 𝑦 ∈ 𝒫 ∪ 𝐴) |
19 | 18 | abssi 4043 | . . 3 ⊢ {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = ∩ 𝑥} ⊆ 𝒫 ∪ 𝐴 |
20 | uniexg 7456 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V) | |
21 | 20 | pwexd 5271 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝒫 ∪ 𝐴 ∈ V) |
22 | ssexg 5218 | . . 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 6783 | 1 ⊢ (𝐴 ∈ 𝑉 → (fi‘𝐴) = {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = ∩ 𝑥}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 {cab 2796 ≠ wne 3013 ∃wrex 3136 Vcvv 3492 ∩ cin 3932 ⊆ wss 3933 ∅c0 4288 𝒫 cpw 4535 ∪ cuni 4830 ∩ cint 4867 ‘cfv 6348 Fincfn 8497 ficfi 8862 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-int 4868 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-fi 8863 |
This theorem is referenced by: elfi 8865 fi0 8872 |
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