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Theorem fbun 21691
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 3813 . . . . 5 (𝑥𝐹𝑥 ∈ (𝐹𝐺))
2 elun2 3814 . . . . 5 (𝑦𝐺𝑦 ∈ (𝐹𝐺))
31, 2anim12i 589 . . . 4 ((𝑥𝐹𝑦𝐺) → (𝑥 ∈ (𝐹𝐺) ∧ 𝑦 ∈ (𝐹𝐺)))
4 fbasssin 21687 . . . . 5 (((𝐹𝐺) ∈ (fBas‘𝑋) ∧ 𝑥 ∈ (𝐹𝐺) ∧ 𝑦 ∈ (𝐹𝐺)) → ∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
543expb 1285 . . . 4 (((𝐹𝐺) ∈ (fBas‘𝑋) ∧ (𝑥 ∈ (𝐹𝐺) ∧ 𝑦 ∈ (𝐹𝐺))) → ∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
63, 5sylan2 490 . . 3 (((𝐹𝐺) ∈ (fBas‘𝑋) ∧ (𝑥𝐹𝑦𝐺)) → ∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
76ralrimivva 3000 . 2 ((𝐹𝐺) ∈ (fBas‘𝑋) → ∀𝑥𝐹𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
8 fbsspw 21683 . . . . . . 7 (𝐹 ∈ (fBas‘𝑋) → 𝐹 ⊆ 𝒫 𝑋)
98adantr 480 . . . . . 6 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → 𝐹 ⊆ 𝒫 𝑋)
10 fbsspw 21683 . . . . . . 7 (𝐺 ∈ (fBas‘𝑋) → 𝐺 ⊆ 𝒫 𝑋)
1110adantl 481 . . . . . 6 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → 𝐺 ⊆ 𝒫 𝑋)
129, 11unssd 3822 . . . . 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 3809 . . . . . . . 8 𝐹 ⊆ (𝐹𝐺)
15 fbasne0 21681 . . . . . . . 8 (𝐹 ∈ (fBas‘𝑋) → 𝐹 ≠ ∅)
16 ssn0 4009 . . . . . . . 8 ((𝐹 ⊆ (𝐹𝐺) ∧ 𝐹 ≠ ∅) → (𝐹𝐺) ≠ ∅)
1714, 15, 16sylancr 696 . . . . . . 7 (𝐹 ∈ (fBas‘𝑋) → (𝐹𝐺) ≠ ∅)
1817adantr 480 . . . . . 6 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (𝐹𝐺) ≠ ∅)
1918a1d 25 . . . . 5 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (∀𝑥𝐹𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) → (𝐹𝐺) ≠ ∅))
20 0nelfb 21682 . . . . . . 7 (𝐹 ∈ (fBas‘𝑋) → ¬ ∅ ∈ 𝐹)
21 0nelfb 21682 . . . . . . 7 (𝐺 ∈ (fBas‘𝑋) → ¬ ∅ ∈ 𝐺)
22 df-nel 2927 . . . . . . . . 9 (∅ ∉ (𝐹𝐺) ↔ ¬ ∅ ∈ (𝐹𝐺))
23 elun 3786 . . . . . . . . . 10 (∅ ∈ (𝐹𝐺) ↔ (∅ ∈ 𝐹 ∨ ∅ ∈ 𝐺))
2423notbii 309 . . . . . . . . 9 (¬ ∅ ∈ (𝐹𝐺) ↔ ¬ (∅ ∈ 𝐹 ∨ ∅ ∈ 𝐺))
25 ioran 510 . . . . . . . . 9 (¬ (∅ ∈ 𝐹 ∨ ∅ ∈ 𝐺) ↔ (¬ ∅ ∈ 𝐹 ∧ ¬ ∅ ∈ 𝐺))
2622, 24, 253bitri 286 . . . . . . . 8 (∅ ∉ (𝐹𝐺) ↔ (¬ ∅ ∈ 𝐹 ∧ ¬ ∅ ∈ 𝐺))
2726biimpri 218 . . . . . . 7 ((¬ ∅ ∈ 𝐹 ∧ ¬ ∅ ∈ 𝐺) → ∅ ∉ (𝐹𝐺))
2820, 21, 27syl2an 493 . . . . . 6 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → ∅ ∉ (𝐹𝐺))
2928a1d 25 . . . . 5 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (∀𝑥𝐹𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) → ∅ ∉ (𝐹𝐺)))
30 fbasssin 21687 . . . . . . . . . . . . 13 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑥𝐹𝑦𝐹) → ∃𝑧𝐹 𝑧 ⊆ (𝑥𝑦))
31 ssrexv 3700 . . . . . . . . . . . . 13 (𝐹 ⊆ (𝐹𝐺) → (∃𝑧𝐹 𝑧 ⊆ (𝑥𝑦) → ∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)))
3214, 30, 31mpsyl 68 . . . . . . . . . . . 12 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑥𝐹𝑦𝐹) → ∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
33323expb 1285 . . . . . . . . . . 11 ((𝐹 ∈ (fBas‘𝑋) ∧ (𝑥𝐹𝑦𝐹)) → ∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
3433ralrimivva 3000 . . . . . . . . . 10 (𝐹 ∈ (fBas‘𝑋) → ∀𝑥𝐹𝑦𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
3534adantr 480 . . . . . . . . 9 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → ∀𝑥𝐹𝑦𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
36 pm3.2 462 . . . . . . . . 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 3093 . . . . . . . . 9 (∀𝑥𝐹 (∀𝑦𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ∧ ∀𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)) ↔ (∀𝑥𝐹𝑦𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ∧ ∀𝑥𝐹𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)))
39 ralun 3828 . . . . . . . . . 10 ((∀𝑦𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ∧ ∀𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)) → ∀𝑦 ∈ (𝐹𝐺)∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
4039ralimi 2981 . . . . . . . . 9 (∀𝑥𝐹 (∀𝑦𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ∧ ∀𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)) → ∀𝑥𝐹𝑦 ∈ (𝐹𝐺)∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
4138, 40sylbir 225 . . . . . . . 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 3127 . . . . . . . . . . . 12 (∀𝑥𝐹𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ↔ ∀𝑦𝐺𝑥𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
44 ineq1 3840 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑤 → (𝑥𝑦) = (𝑤𝑦))
4544sseq2d 3666 . . . . . . . . . . . . . . 15 (𝑥 = 𝑤 → (𝑧 ⊆ (𝑥𝑦) ↔ 𝑧 ⊆ (𝑤𝑦)))
4645rexbidv 3081 . . . . . . . . . . . . . 14 (𝑥 = 𝑤 → (∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ↔ ∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑤𝑦)))
4746cbvralv 3201 . . . . . . . . . . . . 13 (∀𝑥𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ↔ ∀𝑤𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑤𝑦))
4847ralbii 3009 . . . . . . . . . . . 12 (∀𝑦𝐺𝑥𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ↔ ∀𝑦𝐺𝑤𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑤𝑦))
49 ineq2 3841 . . . . . . . . . . . . . . 15 (𝑦 = 𝑥 → (𝑤𝑦) = (𝑤𝑥))
5049sseq2d 3666 . . . . . . . . . . . . . 14 (𝑦 = 𝑥 → (𝑧 ⊆ (𝑤𝑦) ↔ 𝑧 ⊆ (𝑤𝑥)))
5150rexbidv 3081 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → (∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑤𝑦) ↔ ∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑤𝑥)))
52 ineq1 3840 . . . . . . . . . . . . . . . 16 (𝑤 = 𝑦 → (𝑤𝑥) = (𝑦𝑥))
53 incom 3838 . . . . . . . . . . . . . . . 16 (𝑦𝑥) = (𝑥𝑦)
5452, 53syl6eq 2701 . . . . . . . . . . . . . . 15 (𝑤 = 𝑦 → (𝑤𝑥) = (𝑥𝑦))
5554sseq2d 3666 . . . . . . . . . . . . . 14 (𝑤 = 𝑦 → (𝑧 ⊆ (𝑤𝑥) ↔ 𝑧 ⊆ (𝑥𝑦)))
5655rexbidv 3081 . . . . . . . . . . . . 13 (𝑤 = 𝑦 → (∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑤𝑥) ↔ ∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)))
5751, 56cbvral2v 3209 . . . . . . . . . . . 12 (∀𝑦𝐺𝑤𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑤𝑦) ↔ ∀𝑥𝐺𝑦𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
5843, 48, 573bitri 286 . . . . . . . . . . 11 (∀𝑥𝐹𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ↔ ∀𝑥𝐺𝑦𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
5958biimpi 206 . . . . . . . . . 10 (∀𝑥𝐹𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) → ∀𝑥𝐺𝑦𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
60 ssun2 3810 . . . . . . . . . . . . . 14 𝐺 ⊆ (𝐹𝐺)
61 fbasssin 21687 . . . . . . . . . . . . . 14 ((𝐺 ∈ (fBas‘𝑋) ∧ 𝑥𝐺𝑦𝐺) → ∃𝑧𝐺 𝑧 ⊆ (𝑥𝑦))
62 ssrexv 3700 . . . . . . . . . . . . . 14 (𝐺 ⊆ (𝐹𝐺) → (∃𝑧𝐺 𝑧 ⊆ (𝑥𝑦) → ∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)))
6360, 61, 62mpsyl 68 . . . . . . . . . . . . 13 ((𝐺 ∈ (fBas‘𝑋) ∧ 𝑥𝐺𝑦𝐺) → ∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
64633expb 1285 . . . . . . . . . . . 12 ((𝐺 ∈ (fBas‘𝑋) ∧ (𝑥𝐺𝑦𝐺)) → ∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
6564ralrimivva 3000 . . . . . . . . . . 11 (𝐺 ∈ (fBas‘𝑋) → ∀𝑥𝐺𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
6665adantl 481 . . . . . . . . . 10 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → ∀𝑥𝐺𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
6759, 66anim12i 589 . . . . . . . . 9 ((∀𝑥𝐹𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ∧ (𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋))) → (∀𝑥𝐺𝑦𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ∧ ∀𝑥𝐺𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)))
6867expcom 450 . . . . . . . 8 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (∀𝑥𝐹𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) → (∀𝑥𝐺𝑦𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ∧ ∀𝑥𝐺𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))))
69 r19.26 3093 . . . . . . . . 9 (∀𝑥𝐺 (∀𝑦𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ∧ ∀𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)) ↔ (∀𝑥𝐺𝑦𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ∧ ∀𝑥𝐺𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)))
7039ralimi 2981 . . . . . . . . 9 (∀𝑥𝐺 (∀𝑦𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ∧ ∀𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)) → ∀𝑥𝐺𝑦 ∈ (𝐹𝐺)∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
7169, 70sylbir 225 . . . . . . . 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 554 . . . . . 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 3828 . . . . . 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 1262 . . . 4 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (∀𝑥𝐹𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) → ((𝐹𝐺) ≠ ∅ ∧ ∅ ∉ (𝐹𝐺) ∧ ∀𝑥 ∈ (𝐹𝐺)∀𝑦 ∈ (𝐹𝐺)∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))))
7713, 76jcad 554 . . 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 6258 . . . . 5 (𝐹 ∈ (fBas‘𝑋) → 𝑋 ∈ dom fBas)
7978adantr 480 . . . 4 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → 𝑋 ∈ dom fBas)
80 isfbas2 21686 . . . 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 249 . 2 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (∀𝑥𝐹𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) → (𝐹𝐺) ∈ (fBas‘𝑋)))
837, 82impbid2 216 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 196  wo 382  wa 383  w3a 1054  wcel 2030  wne 2823  wnel 2926  wral 2941  wrex 2942  cun 3605  cin 3606  wss 3607  c0 3948  𝒫 cpw 4191  dom cdm 5143  cfv 5926  fBascfbas 19782
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fv 5934  df-fbas 19791
This theorem is referenced by: (None)
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