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Theorem fbun 22364
Description: A necessary and sufficient condition for the union of two filter bases to also be a filter base. (Contributed by Mario Carneiro, 28-Nov-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
Assertion
Ref Expression
fbun ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → ((𝐹𝐺) ∈ (fBas‘𝑋) ↔ ∀𝑥𝐹𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐺   𝑥,𝐹,𝑦,𝑧   𝑥,𝑋,𝑦,𝑧

Proof of Theorem fbun
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 elun1 4156 . . . . 5 (𝑥𝐹𝑥 ∈ (𝐹𝐺))
2 elun2 4157 . . . . 5 (𝑦𝐺𝑦 ∈ (𝐹𝐺))
31, 2anim12i 612 . . . 4 ((𝑥𝐹𝑦𝐺) → (𝑥 ∈ (𝐹𝐺) ∧ 𝑦 ∈ (𝐹𝐺)))
4 fbasssin 22360 . . . . 5 (((𝐹𝐺) ∈ (fBas‘𝑋) ∧ 𝑥 ∈ (𝐹𝐺) ∧ 𝑦 ∈ (𝐹𝐺)) → ∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
543expb 1114 . . . 4 (((𝐹𝐺) ∈ (fBas‘𝑋) ∧ (𝑥 ∈ (𝐹𝐺) ∧ 𝑦 ∈ (𝐹𝐺))) → ∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
63, 5sylan2 592 . . 3 (((𝐹𝐺) ∈ (fBas‘𝑋) ∧ (𝑥𝐹𝑦𝐺)) → ∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
76ralrimivva 3196 . 2 ((𝐹𝐺) ∈ (fBas‘𝑋) → ∀𝑥𝐹𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
8 fbsspw 22356 . . . . . . 7 (𝐹 ∈ (fBas‘𝑋) → 𝐹 ⊆ 𝒫 𝑋)
98adantr 481 . . . . . 6 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → 𝐹 ⊆ 𝒫 𝑋)
10 fbsspw 22356 . . . . . . 7 (𝐺 ∈ (fBas‘𝑋) → 𝐺 ⊆ 𝒫 𝑋)
1110adantl 482 . . . . . 6 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → 𝐺 ⊆ 𝒫 𝑋)
129, 11unssd 4166 . . . . 5 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (𝐹𝐺) ⊆ 𝒫 𝑋)
1312a1d 25 . . . 4 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (∀𝑥𝐹𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) → (𝐹𝐺) ⊆ 𝒫 𝑋))
14 ssun1 4152 . . . . . . . 8 𝐹 ⊆ (𝐹𝐺)
15 fbasne0 22354 . . . . . . . 8 (𝐹 ∈ (fBas‘𝑋) → 𝐹 ≠ ∅)
16 ssn0 4358 . . . . . . . 8 ((𝐹 ⊆ (𝐹𝐺) ∧ 𝐹 ≠ ∅) → (𝐹𝐺) ≠ ∅)
1714, 15, 16sylancr 587 . . . . . . 7 (𝐹 ∈ (fBas‘𝑋) → (𝐹𝐺) ≠ ∅)
1817adantr 481 . . . . . 6 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (𝐹𝐺) ≠ ∅)
1918a1d 25 . . . . 5 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (∀𝑥𝐹𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) → (𝐹𝐺) ≠ ∅))
20 0nelfb 22355 . . . . . . 7 (𝐹 ∈ (fBas‘𝑋) → ¬ ∅ ∈ 𝐹)
21 0nelfb 22355 . . . . . . 7 (𝐺 ∈ (fBas‘𝑋) → ¬ ∅ ∈ 𝐺)
22 df-nel 3129 . . . . . . . . 9 (∅ ∉ (𝐹𝐺) ↔ ¬ ∅ ∈ (𝐹𝐺))
23 elun 4129 . . . . . . . . . 10 (∅ ∈ (𝐹𝐺) ↔ (∅ ∈ 𝐹 ∨ ∅ ∈ 𝐺))
2423notbii 321 . . . . . . . . 9 (¬ ∅ ∈ (𝐹𝐺) ↔ ¬ (∅ ∈ 𝐹 ∨ ∅ ∈ 𝐺))
25 ioran 979 . . . . . . . . 9 (¬ (∅ ∈ 𝐹 ∨ ∅ ∈ 𝐺) ↔ (¬ ∅ ∈ 𝐹 ∧ ¬ ∅ ∈ 𝐺))
2622, 24, 253bitri 298 . . . . . . . 8 (∅ ∉ (𝐹𝐺) ↔ (¬ ∅ ∈ 𝐹 ∧ ¬ ∅ ∈ 𝐺))
2726biimpri 229 . . . . . . 7 ((¬ ∅ ∈ 𝐹 ∧ ¬ ∅ ∈ 𝐺) → ∅ ∉ (𝐹𝐺))
2820, 21, 27syl2an 595 . . . . . 6 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → ∅ ∉ (𝐹𝐺))
2928a1d 25 . . . . 5 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (∀𝑥𝐹𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) → ∅ ∉ (𝐹𝐺)))
30 fbasssin 22360 . . . . . . . . . . . . 13 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑥𝐹𝑦𝐹) → ∃𝑧𝐹 𝑧 ⊆ (𝑥𝑦))
31 ssrexv 4038 . . . . . . . . . . . . 13 (𝐹 ⊆ (𝐹𝐺) → (∃𝑧𝐹 𝑧 ⊆ (𝑥𝑦) → ∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)))
3214, 30, 31mpsyl 68 . . . . . . . . . . . 12 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑥𝐹𝑦𝐹) → ∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
33323expb 1114 . . . . . . . . . . 11 ((𝐹 ∈ (fBas‘𝑋) ∧ (𝑥𝐹𝑦𝐹)) → ∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
3433ralrimivva 3196 . . . . . . . . . 10 (𝐹 ∈ (fBas‘𝑋) → ∀𝑥𝐹𝑦𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
3534adantr 481 . . . . . . . . 9 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → ∀𝑥𝐹𝑦𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
36 pm3.2 470 . . . . . . . . 9 (∀𝑥𝐹𝑦𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) → (∀𝑥𝐹𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) → (∀𝑥𝐹𝑦𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ∧ ∀𝑥𝐹𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))))
3735, 36syl 17 . . . . . . . 8 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (∀𝑥𝐹𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) → (∀𝑥𝐹𝑦𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ∧ ∀𝑥𝐹𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))))
38 r19.26 3175 . . . . . . . . 9 (∀𝑥𝐹 (∀𝑦𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ∧ ∀𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)) ↔ (∀𝑥𝐹𝑦𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ∧ ∀𝑥𝐹𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)))
39 ralun 4172 . . . . . . . . . 10 ((∀𝑦𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ∧ ∀𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)) → ∀𝑦 ∈ (𝐹𝐺)∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
4039ralimi 3165 . . . . . . . . 9 (∀𝑥𝐹 (∀𝑦𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ∧ ∀𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)) → ∀𝑥𝐹𝑦 ∈ (𝐹𝐺)∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
4138, 40sylbir 236 . . . . . . . 8 ((∀𝑥𝐹𝑦𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ∧ ∀𝑥𝐹𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)) → ∀𝑥𝐹𝑦 ∈ (𝐹𝐺)∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
4237, 41syl6 35 . . . . . . 7 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (∀𝑥𝐹𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) → ∀𝑥𝐹𝑦 ∈ (𝐹𝐺)∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)))
43 ralcom 3359 . . . . . . . . . . . 12 (∀𝑥𝐹𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ↔ ∀𝑦𝐺𝑥𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
44 ineq1 4185 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑤 → (𝑥𝑦) = (𝑤𝑦))
4544sseq2d 4003 . . . . . . . . . . . . . . 15 (𝑥 = 𝑤 → (𝑧 ⊆ (𝑥𝑦) ↔ 𝑧 ⊆ (𝑤𝑦)))
4645rexbidv 3302 . . . . . . . . . . . . . 14 (𝑥 = 𝑤 → (∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ↔ ∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑤𝑦)))
4746cbvralvw 3455 . . . . . . . . . . . . 13 (∀𝑥𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ↔ ∀𝑤𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑤𝑦))
4847ralbii 3170 . . . . . . . . . . . 12 (∀𝑦𝐺𝑥𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ↔ ∀𝑦𝐺𝑤𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑤𝑦))
49 ineq2 4187 . . . . . . . . . . . . . . 15 (𝑦 = 𝑥 → (𝑤𝑦) = (𝑤𝑥))
5049sseq2d 4003 . . . . . . . . . . . . . 14 (𝑦 = 𝑥 → (𝑧 ⊆ (𝑤𝑦) ↔ 𝑧 ⊆ (𝑤𝑥)))
5150rexbidv 3302 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → (∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑤𝑦) ↔ ∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑤𝑥)))
52 ineq1 4185 . . . . . . . . . . . . . . . 16 (𝑤 = 𝑦 → (𝑤𝑥) = (𝑦𝑥))
53 incom 4182 . . . . . . . . . . . . . . . 16 (𝑦𝑥) = (𝑥𝑦)
5452, 53syl6eq 2877 . . . . . . . . . . . . . . 15 (𝑤 = 𝑦 → (𝑤𝑥) = (𝑥𝑦))
5554sseq2d 4003 . . . . . . . . . . . . . 14 (𝑤 = 𝑦 → (𝑧 ⊆ (𝑤𝑥) ↔ 𝑧 ⊆ (𝑥𝑦)))
5655rexbidv 3302 . . . . . . . . . . . . 13 (𝑤 = 𝑦 → (∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑤𝑥) ↔ ∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)))
5751, 56cbvral2vw 3467 . . . . . . . . . . . 12 (∀𝑦𝐺𝑤𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑤𝑦) ↔ ∀𝑥𝐺𝑦𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
5843, 48, 573bitri 298 . . . . . . . . . . 11 (∀𝑥𝐹𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ↔ ∀𝑥𝐺𝑦𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
5958biimpi 217 . . . . . . . . . 10 (∀𝑥𝐹𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) → ∀𝑥𝐺𝑦𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
60 ssun2 4153 . . . . . . . . . . . . . 14 𝐺 ⊆ (𝐹𝐺)
61 fbasssin 22360 . . . . . . . . . . . . . 14 ((𝐺 ∈ (fBas‘𝑋) ∧ 𝑥𝐺𝑦𝐺) → ∃𝑧𝐺 𝑧 ⊆ (𝑥𝑦))
62 ssrexv 4038 . . . . . . . . . . . . . 14 (𝐺 ⊆ (𝐹𝐺) → (∃𝑧𝐺 𝑧 ⊆ (𝑥𝑦) → ∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)))
6360, 61, 62mpsyl 68 . . . . . . . . . . . . 13 ((𝐺 ∈ (fBas‘𝑋) ∧ 𝑥𝐺𝑦𝐺) → ∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
64633expb 1114 . . . . . . . . . . . 12 ((𝐺 ∈ (fBas‘𝑋) ∧ (𝑥𝐺𝑦𝐺)) → ∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
6564ralrimivva 3196 . . . . . . . . . . 11 (𝐺 ∈ (fBas‘𝑋) → ∀𝑥𝐺𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
6665adantl 482 . . . . . . . . . 10 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → ∀𝑥𝐺𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
6759, 66anim12i 612 . . . . . . . . 9 ((∀𝑥𝐹𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ∧ (𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋))) → (∀𝑥𝐺𝑦𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ∧ ∀𝑥𝐺𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)))
6867expcom 414 . . . . . . . 8 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (∀𝑥𝐹𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) → (∀𝑥𝐺𝑦𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ∧ ∀𝑥𝐺𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))))
69 r19.26 3175 . . . . . . . . 9 (∀𝑥𝐺 (∀𝑦𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ∧ ∀𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)) ↔ (∀𝑥𝐺𝑦𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ∧ ∀𝑥𝐺𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)))
7039ralimi 3165 . . . . . . . . 9 (∀𝑥𝐺 (∀𝑦𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ∧ ∀𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)) → ∀𝑥𝐺𝑦 ∈ (𝐹𝐺)∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
7169, 70sylbir 236 . . . . . . . 8 ((∀𝑥𝐺𝑦𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ∧ ∀𝑥𝐺𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)) → ∀𝑥𝐺𝑦 ∈ (𝐹𝐺)∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
7268, 71syl6 35 . . . . . . 7 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (∀𝑥𝐹𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) → ∀𝑥𝐺𝑦 ∈ (𝐹𝐺)∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)))
7342, 72jcad 513 . . . . . 6 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (∀𝑥𝐹𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) → (∀𝑥𝐹𝑦 ∈ (𝐹𝐺)∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ∧ ∀𝑥𝐺𝑦 ∈ (𝐹𝐺)∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))))
74 ralun 4172 . . . . . 6 ((∀𝑥𝐹𝑦 ∈ (𝐹𝐺)∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ∧ ∀𝑥𝐺𝑦 ∈ (𝐹𝐺)∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)) → ∀𝑥 ∈ (𝐹𝐺)∀𝑦 ∈ (𝐹𝐺)∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
7573, 74syl6 35 . . . . 5 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (∀𝑥𝐹𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) → ∀𝑥 ∈ (𝐹𝐺)∀𝑦 ∈ (𝐹𝐺)∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)))
7619, 29, 753jcad 1123 . . . 4 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (∀𝑥𝐹𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) → ((𝐹𝐺) ≠ ∅ ∧ ∅ ∉ (𝐹𝐺) ∧ ∀𝑥 ∈ (𝐹𝐺)∀𝑦 ∈ (𝐹𝐺)∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))))
7713, 76jcad 513 . . 3 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (∀𝑥𝐹𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) → ((𝐹𝐺) ⊆ 𝒫 𝑋 ∧ ((𝐹𝐺) ≠ ∅ ∧ ∅ ∉ (𝐹𝐺) ∧ ∀𝑥 ∈ (𝐹𝐺)∀𝑦 ∈ (𝐹𝐺)∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)))))
78 elfvdm 6699 . . . . 5 (𝐹 ∈ (fBas‘𝑋) → 𝑋 ∈ dom fBas)
7978adantr 481 . . . 4 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → 𝑋 ∈ dom fBas)
80 isfbas2 22359 . . . 4 (𝑋 ∈ dom fBas → ((𝐹𝐺) ∈ (fBas‘𝑋) ↔ ((𝐹𝐺) ⊆ 𝒫 𝑋 ∧ ((𝐹𝐺) ≠ ∅ ∧ ∅ ∉ (𝐹𝐺) ∧ ∀𝑥 ∈ (𝐹𝐺)∀𝑦 ∈ (𝐹𝐺)∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)))))
8179, 80syl 17 . . 3 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → ((𝐹𝐺) ∈ (fBas‘𝑋) ↔ ((𝐹𝐺) ⊆ 𝒫 𝑋 ∧ ((𝐹𝐺) ≠ ∅ ∧ ∅ ∉ (𝐹𝐺) ∧ ∀𝑥 ∈ (𝐹𝐺)∀𝑦 ∈ (𝐹𝐺)∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)))))
8277, 81sylibrd 260 . 2 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (∀𝑥𝐹𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) → (𝐹𝐺) ∈ (fBas‘𝑋)))
837, 82impbid2 227 1 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → ((𝐹𝐺) ∈ (fBas‘𝑋) ↔ ∀𝑥𝐹𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 843  w3a 1081  wcel 2107  wne 3021  wnel 3128  wral 3143  wrex 3144  cun 3938  cin 3939  wss 3940  c0 4295  𝒫 cpw 4542  dom cdm 5554  cfv 6352  fBascfbas 20449
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fv 6360  df-fbas 20458
This theorem is referenced by: (None)
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