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Mirrors > Home > MPE Home > Th. List > fbssint | Structured version Visualization version GIF version |
Description: A filter base contains subsets of its finite intersections. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Stefan O'Rear, 28-Jul-2015.) |
Ref | Expression |
---|---|
fbssint | ⊢ ((𝐹 ∈ (fBas‘𝐵) ∧ 𝐴 ⊆ 𝐹 ∧ 𝐴 ∈ Fin) → ∃𝑥 ∈ 𝐹 𝑥 ⊆ ∩ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fbasne0 22438 | . . . . . 6 ⊢ (𝐹 ∈ (fBas‘𝐵) → 𝐹 ≠ ∅) | |
2 | n0 4310 | . . . . . 6 ⊢ (𝐹 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐹) | |
3 | 1, 2 | sylib 220 | . . . . 5 ⊢ (𝐹 ∈ (fBas‘𝐵) → ∃𝑥 𝑥 ∈ 𝐹) |
4 | ssv 3991 | . . . . . . . 8 ⊢ 𝑥 ⊆ V | |
5 | 4 | jctr 527 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐹 → (𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ V)) |
6 | 5 | eximi 1835 | . . . . . 6 ⊢ (∃𝑥 𝑥 ∈ 𝐹 → ∃𝑥(𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ V)) |
7 | df-rex 3144 | . . . . . 6 ⊢ (∃𝑥 ∈ 𝐹 𝑥 ⊆ V ↔ ∃𝑥(𝑥 ∈ 𝐹 ∧ 𝑥 ⊆ V)) | |
8 | 6, 7 | sylibr 236 | . . . . 5 ⊢ (∃𝑥 𝑥 ∈ 𝐹 → ∃𝑥 ∈ 𝐹 𝑥 ⊆ V) |
9 | 3, 8 | syl 17 | . . . 4 ⊢ (𝐹 ∈ (fBas‘𝐵) → ∃𝑥 ∈ 𝐹 𝑥 ⊆ V) |
10 | inteq 4879 | . . . . . . 7 ⊢ (𝐴 = ∅ → ∩ 𝐴 = ∩ ∅) | |
11 | int0 4890 | . . . . . . 7 ⊢ ∩ ∅ = V | |
12 | 10, 11 | syl6eq 2872 | . . . . . 6 ⊢ (𝐴 = ∅ → ∩ 𝐴 = V) |
13 | 12 | sseq2d 3999 | . . . . 5 ⊢ (𝐴 = ∅ → (𝑥 ⊆ ∩ 𝐴 ↔ 𝑥 ⊆ V)) |
14 | 13 | rexbidv 3297 | . . . 4 ⊢ (𝐴 = ∅ → (∃𝑥 ∈ 𝐹 𝑥 ⊆ ∩ 𝐴 ↔ ∃𝑥 ∈ 𝐹 𝑥 ⊆ V)) |
15 | 9, 14 | syl5ibrcom 249 | . . 3 ⊢ (𝐹 ∈ (fBas‘𝐵) → (𝐴 = ∅ → ∃𝑥 ∈ 𝐹 𝑥 ⊆ ∩ 𝐴)) |
16 | 15 | 3ad2ant1 1129 | . 2 ⊢ ((𝐹 ∈ (fBas‘𝐵) ∧ 𝐴 ⊆ 𝐹 ∧ 𝐴 ∈ Fin) → (𝐴 = ∅ → ∃𝑥 ∈ 𝐹 𝑥 ⊆ ∩ 𝐴)) |
17 | simpl1 1187 | . . . 4 ⊢ (((𝐹 ∈ (fBas‘𝐵) ∧ 𝐴 ⊆ 𝐹 ∧ 𝐴 ∈ Fin) ∧ 𝐴 ≠ ∅) → 𝐹 ∈ (fBas‘𝐵)) | |
18 | simpl2 1188 | . . . . 5 ⊢ (((𝐹 ∈ (fBas‘𝐵) ∧ 𝐴 ⊆ 𝐹 ∧ 𝐴 ∈ Fin) ∧ 𝐴 ≠ ∅) → 𝐴 ⊆ 𝐹) | |
19 | simpr 487 | . . . . 5 ⊢ (((𝐹 ∈ (fBas‘𝐵) ∧ 𝐴 ⊆ 𝐹 ∧ 𝐴 ∈ Fin) ∧ 𝐴 ≠ ∅) → 𝐴 ≠ ∅) | |
20 | simpl3 1189 | . . . . 5 ⊢ (((𝐹 ∈ (fBas‘𝐵) ∧ 𝐴 ⊆ 𝐹 ∧ 𝐴 ∈ Fin) ∧ 𝐴 ≠ ∅) → 𝐴 ∈ Fin) | |
21 | elfir 8879 | . . . . 5 ⊢ ((𝐹 ∈ (fBas‘𝐵) ∧ (𝐴 ⊆ 𝐹 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → ∩ 𝐴 ∈ (fi‘𝐹)) | |
22 | 17, 18, 19, 20, 21 | syl13anc 1368 | . . . 4 ⊢ (((𝐹 ∈ (fBas‘𝐵) ∧ 𝐴 ⊆ 𝐹 ∧ 𝐴 ∈ Fin) ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ (fi‘𝐹)) |
23 | fbssfi 22445 | . . . 4 ⊢ ((𝐹 ∈ (fBas‘𝐵) ∧ ∩ 𝐴 ∈ (fi‘𝐹)) → ∃𝑥 ∈ 𝐹 𝑥 ⊆ ∩ 𝐴) | |
24 | 17, 22, 23 | syl2anc 586 | . . 3 ⊢ (((𝐹 ∈ (fBas‘𝐵) ∧ 𝐴 ⊆ 𝐹 ∧ 𝐴 ∈ Fin) ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐹 𝑥 ⊆ ∩ 𝐴) |
25 | 24 | ex 415 | . 2 ⊢ ((𝐹 ∈ (fBas‘𝐵) ∧ 𝐴 ⊆ 𝐹 ∧ 𝐴 ∈ Fin) → (𝐴 ≠ ∅ → ∃𝑥 ∈ 𝐹 𝑥 ⊆ ∩ 𝐴)) |
26 | 16, 25 | pm2.61dne 3103 | 1 ⊢ ((𝐹 ∈ (fBas‘𝐵) ∧ 𝐴 ⊆ 𝐹 ∧ 𝐴 ∈ Fin) → ∃𝑥 ∈ 𝐹 𝑥 ⊆ ∩ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∃wex 1780 ∈ wcel 2114 ≠ wne 3016 ∃wrex 3139 Vcvv 3494 ⊆ wss 3936 ∅c0 4291 ∩ cint 4876 ‘cfv 6355 Fincfn 8509 ficfi 8874 fBascfbas 20533 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-fin 8513 df-fi 8875 df-fbas 20542 |
This theorem is referenced by: fbasfip 22476 |
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