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Theorem fbun 23097
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 4123 . . . . 5 (𝑥𝐹𝑥 ∈ (𝐹𝐺))
2 elun2 4124 . . . . 5 (𝑦𝐺𝑦 ∈ (𝐹𝐺))
31, 2anim12i 613 . . . 4 ((𝑥𝐹𝑦𝐺) → (𝑥 ∈ (𝐹𝐺) ∧ 𝑦 ∈ (𝐹𝐺)))
4 fbasssin 23093 . . . . 5 (((𝐹𝐺) ∈ (fBas‘𝑋) ∧ 𝑥 ∈ (𝐹𝐺) ∧ 𝑦 ∈ (𝐹𝐺)) → ∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
543expb 1119 . . . 4 (((𝐹𝐺) ∈ (fBas‘𝑋) ∧ (𝑥 ∈ (𝐹𝐺) ∧ 𝑦 ∈ (𝐹𝐺))) → ∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
63, 5sylan2 593 . . 3 (((𝐹𝐺) ∈ (fBas‘𝑋) ∧ (𝑥𝐹𝑦𝐺)) → ∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
76ralrimivva 3193 . 2 ((𝐹𝐺) ∈ (fBas‘𝑋) → ∀𝑥𝐹𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
8 fbsspw 23089 . . . . . . 7 (𝐹 ∈ (fBas‘𝑋) → 𝐹 ⊆ 𝒫 𝑋)
98adantr 481 . . . . . 6 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → 𝐹 ⊆ 𝒫 𝑋)
10 fbsspw 23089 . . . . . . 7 (𝐺 ∈ (fBas‘𝑋) → 𝐺 ⊆ 𝒫 𝑋)
1110adantl 482 . . . . . 6 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → 𝐺 ⊆ 𝒫 𝑋)
129, 11unssd 4133 . . . . 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 4119 . . . . . . . 8 𝐹 ⊆ (𝐹𝐺)
15 fbasne0 23087 . . . . . . . 8 (𝐹 ∈ (fBas‘𝑋) → 𝐹 ≠ ∅)
16 ssn0 4347 . . . . . . . 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 23088 . . . . . . 7 (𝐹 ∈ (fBas‘𝑋) → ¬ ∅ ∈ 𝐹)
21 0nelfb 23088 . . . . . . 7 (𝐺 ∈ (fBas‘𝑋) → ¬ ∅ ∈ 𝐺)
22 df-nel 3047 . . . . . . . . 9 (∅ ∉ (𝐹𝐺) ↔ ¬ ∅ ∈ (𝐹𝐺))
23 elun 4095 . . . . . . . . . 10 (∅ ∈ (𝐹𝐺) ↔ (∅ ∈ 𝐹 ∨ ∅ ∈ 𝐺))
2423notbii 319 . . . . . . . . 9 (¬ ∅ ∈ (𝐹𝐺) ↔ ¬ (∅ ∈ 𝐹 ∨ ∅ ∈ 𝐺))
25 ioran 981 . . . . . . . . 9 (¬ (∅ ∈ 𝐹 ∨ ∅ ∈ 𝐺) ↔ (¬ ∅ ∈ 𝐹 ∧ ¬ ∅ ∈ 𝐺))
2622, 24, 253bitri 296 . . . . . . . 8 (∅ ∉ (𝐹𝐺) ↔ (¬ ∅ ∈ 𝐹 ∧ ¬ ∅ ∈ 𝐺))
2726biimpri 227 . . . . . . 7 ((¬ ∅ ∈ 𝐹 ∧ ¬ ∅ ∈ 𝐺) → ∅ ∉ (𝐹𝐺))
2820, 21, 27syl2an 596 . . . . . 6 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → ∅ ∉ (𝐹𝐺))
2928a1d 25 . . . . 5 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (∀𝑥𝐹𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) → ∅ ∉ (𝐹𝐺)))
30 fbasssin 23093 . . . . . . . . . . . . 13 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑥𝐹𝑦𝐹) → ∃𝑧𝐹 𝑧 ⊆ (𝑥𝑦))
31 ssrexv 3999 . . . . . . . . . . . . 13 (𝐹 ⊆ (𝐹𝐺) → (∃𝑧𝐹 𝑧 ⊆ (𝑥𝑦) → ∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)))
3214, 30, 31mpsyl 68 . . . . . . . . . . . 12 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑥𝐹𝑦𝐹) → ∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
33323expb 1119 . . . . . . . . . . 11 ((𝐹 ∈ (fBas‘𝑋) ∧ (𝑥𝐹𝑦𝐹)) → ∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
3433ralrimivva 3193 . . . . . . . . . 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 3110 . . . . . . . . 9 (∀𝑥𝐹 (∀𝑦𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ∧ ∀𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)) ↔ (∀𝑥𝐹𝑦𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ∧ ∀𝑥𝐹𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)))
39 ralun 4139 . . . . . . . . . 10 ((∀𝑦𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ∧ ∀𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)) → ∀𝑦 ∈ (𝐹𝐺)∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
4039ralimi 3082 . . . . . . . . 9 (∀𝑥𝐹 (∀𝑦𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ∧ ∀𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)) → ∀𝑥𝐹𝑦 ∈ (𝐹𝐺)∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
4138, 40sylbir 234 . . . . . . . 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 3268 . . . . . . . . . . . 12 (∀𝑥𝐹𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ↔ ∀𝑦𝐺𝑥𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
44 ineq1 4152 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑤 → (𝑥𝑦) = (𝑤𝑦))
4544sseq2d 3964 . . . . . . . . . . . . . . 15 (𝑥 = 𝑤 → (𝑧 ⊆ (𝑥𝑦) ↔ 𝑧 ⊆ (𝑤𝑦)))
4645rexbidv 3171 . . . . . . . . . . . . . 14 (𝑥 = 𝑤 → (∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ↔ ∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑤𝑦)))
4746cbvralvw 3221 . . . . . . . . . . . . 13 (∀𝑥𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ↔ ∀𝑤𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑤𝑦))
4847ralbii 3092 . . . . . . . . . . . 12 (∀𝑦𝐺𝑥𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ↔ ∀𝑦𝐺𝑤𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑤𝑦))
49 ineq2 4153 . . . . . . . . . . . . . . 15 (𝑦 = 𝑥 → (𝑤𝑦) = (𝑤𝑥))
5049sseq2d 3964 . . . . . . . . . . . . . 14 (𝑦 = 𝑥 → (𝑧 ⊆ (𝑤𝑦) ↔ 𝑧 ⊆ (𝑤𝑥)))
5150rexbidv 3171 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → (∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑤𝑦) ↔ ∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑤𝑥)))
52 ineq1 4152 . . . . . . . . . . . . . . . 16 (𝑤 = 𝑦 → (𝑤𝑥) = (𝑦𝑥))
53 incom 4148 . . . . . . . . . . . . . . . 16 (𝑦𝑥) = (𝑥𝑦)
5452, 53eqtrdi 2792 . . . . . . . . . . . . . . 15 (𝑤 = 𝑦 → (𝑤𝑥) = (𝑥𝑦))
5554sseq2d 3964 . . . . . . . . . . . . . 14 (𝑤 = 𝑦 → (𝑧 ⊆ (𝑤𝑥) ↔ 𝑧 ⊆ (𝑥𝑦)))
5655rexbidv 3171 . . . . . . . . . . . . 13 (𝑤 = 𝑦 → (∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑤𝑥) ↔ ∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)))
5751, 56cbvral2vw 3223 . . . . . . . . . . . 12 (∀𝑦𝐺𝑤𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑤𝑦) ↔ ∀𝑥𝐺𝑦𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
5843, 48, 573bitri 296 . . . . . . . . . . 11 (∀𝑥𝐹𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ↔ ∀𝑥𝐺𝑦𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
5958biimpi 215 . . . . . . . . . 10 (∀𝑥𝐹𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) → ∀𝑥𝐺𝑦𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
60 ssun2 4120 . . . . . . . . . . . . . 14 𝐺 ⊆ (𝐹𝐺)
61 fbasssin 23093 . . . . . . . . . . . . . 14 ((𝐺 ∈ (fBas‘𝑋) ∧ 𝑥𝐺𝑦𝐺) → ∃𝑧𝐺 𝑧 ⊆ (𝑥𝑦))
62 ssrexv 3999 . . . . . . . . . . . . . 14 (𝐺 ⊆ (𝐹𝐺) → (∃𝑧𝐺 𝑧 ⊆ (𝑥𝑦) → ∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)))
6360, 61, 62mpsyl 68 . . . . . . . . . . . . 13 ((𝐺 ∈ (fBas‘𝑋) ∧ 𝑥𝐺𝑦𝐺) → ∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
64633expb 1119 . . . . . . . . . . . 12 ((𝐺 ∈ (fBas‘𝑋) ∧ (𝑥𝐺𝑦𝐺)) → ∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
6564ralrimivva 3193 . . . . . . . . . . 11 (𝐺 ∈ (fBas‘𝑋) → ∀𝑥𝐺𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
6665adantl 482 . . . . . . . . . 10 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → ∀𝑥𝐺𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
6759, 66anim12i 613 . . . . . . . . 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 3110 . . . . . . . . 9 (∀𝑥𝐺 (∀𝑦𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ∧ ∀𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)) ↔ (∀𝑥𝐺𝑦𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ∧ ∀𝑥𝐺𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)))
7039ralimi 3082 . . . . . . . . 9 (∀𝑥𝐺 (∀𝑦𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ∧ ∀𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)) → ∀𝑥𝐺𝑦 ∈ (𝐹𝐺)∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
7169, 70sylbir 234 . . . . . . . 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 4139 . . . . . 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 1128 . . . 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 6862 . . . . 5 (𝐹 ∈ (fBas‘𝑋) → 𝑋 ∈ dom fBas)
7978adantr 481 . . . 4 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → 𝑋 ∈ dom fBas)
80 isfbas2 23092 . . . 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 258 . 2 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (∀𝑥𝐹𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) → (𝐹𝐺) ∈ (fBas‘𝑋)))
837, 82impbid2 225 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 205  wa 396  wo 844  w3a 1086  wcel 2105  wne 2940  wnel 3046  wral 3061  wrex 3070  cun 3896  cin 3897  wss 3898  c0 4269  𝒫 cpw 4547  dom cdm 5620  cfv 6479  fBascfbas 20691
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6431  df-fun 6481  df-fv 6487  df-fbas 20700
This theorem is referenced by: (None)
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