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Theorem fbun 21454
 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 3742 . . . . 5 (𝑥𝐹𝑥 ∈ (𝐹𝐺))
2 elun2 3743 . . . . 5 (𝑦𝐺𝑦 ∈ (𝐹𝐺))
31, 2anim12i 588 . . . 4 ((𝑥𝐹𝑦𝐺) → (𝑥 ∈ (𝐹𝐺) ∧ 𝑦 ∈ (𝐹𝐺)))
4 fbasssin 21450 . . . . 5 (((𝐹𝐺) ∈ (fBas‘𝑋) ∧ 𝑥 ∈ (𝐹𝐺) ∧ 𝑦 ∈ (𝐹𝐺)) → ∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
543expb 1258 . . . 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 2954 . 2 ((𝐹𝐺) ∈ (fBas‘𝑋) → ∀𝑥𝐹𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
8 fbsspw 21446 . . . . . . 7 (𝐹 ∈ (fBas‘𝑋) → 𝐹 ⊆ 𝒫 𝑋)
98adantr 480 . . . . . 6 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → 𝐹 ⊆ 𝒫 𝑋)
10 fbsspw 21446 . . . . . . 7 (𝐺 ∈ (fBas‘𝑋) → 𝐺 ⊆ 𝒫 𝑋)
1110adantl 481 . . . . . 6 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → 𝐺 ⊆ 𝒫 𝑋)
129, 11unssd 3751 . . . . 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 3738 . . . . . . . 8 𝐹 ⊆ (𝐹𝐺)
15 fbasne0 21444 . . . . . . . 8 (𝐹 ∈ (fBas‘𝑋) → 𝐹 ≠ ∅)
16 ssn0 3928 . . . . . . . 8 ((𝐹 ⊆ (𝐹𝐺) ∧ 𝐹 ≠ ∅) → (𝐹𝐺) ≠ ∅)
1714, 15, 16sylancr 694 . . . . . . 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 21445 . . . . . . 7 (𝐹 ∈ (fBas‘𝑋) → ¬ ∅ ∈ 𝐹)
21 0nelfb 21445 . . . . . . 7 (𝐺 ∈ (fBas‘𝑋) → ¬ ∅ ∈ 𝐺)
22 df-nel 2783 . . . . . . . . 9 (∅ ∉ (𝐹𝐺) ↔ ¬ ∅ ∈ (𝐹𝐺))
23 elun 3715 . . . . . . . . . 10 (∅ ∈ (𝐹𝐺) ↔ (∅ ∈ 𝐹 ∨ ∅ ∈ 𝐺))
2423notbii 309 . . . . . . . . 9 (¬ ∅ ∈ (𝐹𝐺) ↔ ¬ (∅ ∈ 𝐹 ∨ ∅ ∈ 𝐺))
25 ioran 510 . . . . . . . . 9 (¬ (∅ ∈ 𝐹 ∨ ∅ ∈ 𝐺) ↔ (¬ ∅ ∈ 𝐹 ∧ ¬ ∅ ∈ 𝐺))
2622, 24, 253bitri 285 . . . . . . . 8 (∅ ∉ (𝐹𝐺) ↔ (¬ ∅ ∈ 𝐹 ∧ ¬ ∅ ∈ 𝐺))
2726biimpri 217 . . . . . . 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 21450 . . . . . . . . . . . . 13 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑥𝐹𝑦𝐹) → ∃𝑧𝐹 𝑧 ⊆ (𝑥𝑦))
31 ssrexv 3630 . . . . . . . . . . . . 13 (𝐹 ⊆ (𝐹𝐺) → (∃𝑧𝐹 𝑧 ⊆ (𝑥𝑦) → ∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)))
3214, 30, 31mpsyl 66 . . . . . . . . . . . 12 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑥𝐹𝑦𝐹) → ∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
33323expb 1258 . . . . . . . . . . 11 ((𝐹 ∈ (fBas‘𝑋) ∧ (𝑥𝐹𝑦𝐹)) → ∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
3433ralrimivva 2954 . . . . . . . . . 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 3046 . . . . . . . . 9 (∀𝑥𝐹 (∀𝑦𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ∧ ∀𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)) ↔ (∀𝑥𝐹𝑦𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ∧ ∀𝑥𝐹𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)))
39 ralun 3757 . . . . . . . . . 10 ((∀𝑦𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ∧ ∀𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)) → ∀𝑦 ∈ (𝐹𝐺)∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
4039ralimi 2936 . . . . . . . . 9 (∀𝑥𝐹 (∀𝑦𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ∧ ∀𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)) → ∀𝑥𝐹𝑦 ∈ (𝐹𝐺)∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
4138, 40sylbir 224 . . . . . . . 8 ((∀𝑥𝐹𝑦𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ∧ ∀𝑥𝐹𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)) → ∀𝑥𝐹𝑦 ∈ (𝐹𝐺)∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
4237, 41syl6 34 . . . . . . 7 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (∀𝑥𝐹𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) → ∀𝑥𝐹𝑦 ∈ (𝐹𝐺)∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)))
43 ralcom 3079 . . . . . . . . . . . 12 (∀𝑥𝐹𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ↔ ∀𝑦𝐺𝑥𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
44 ineq1 3769 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑤 → (𝑥𝑦) = (𝑤𝑦))
4544sseq2d 3596 . . . . . . . . . . . . . . 15 (𝑥 = 𝑤 → (𝑧 ⊆ (𝑥𝑦) ↔ 𝑧 ⊆ (𝑤𝑦)))
4645rexbidv 3034 . . . . . . . . . . . . . 14 (𝑥 = 𝑤 → (∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ↔ ∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑤𝑦)))
4746cbvralv 3147 . . . . . . . . . . . . 13 (∀𝑥𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ↔ ∀𝑤𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑤𝑦))
4847ralbii 2963 . . . . . . . . . . . 12 (∀𝑦𝐺𝑥𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ↔ ∀𝑦𝐺𝑤𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑤𝑦))
49 ineq2 3770 . . . . . . . . . . . . . . 15 (𝑦 = 𝑥 → (𝑤𝑦) = (𝑤𝑥))
5049sseq2d 3596 . . . . . . . . . . . . . 14 (𝑦 = 𝑥 → (𝑧 ⊆ (𝑤𝑦) ↔ 𝑧 ⊆ (𝑤𝑥)))
5150rexbidv 3034 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → (∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑤𝑦) ↔ ∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑤𝑥)))
52 ineq1 3769 . . . . . . . . . . . . . . . 16 (𝑤 = 𝑦 → (𝑤𝑥) = (𝑦𝑥))
53 incom 3767 . . . . . . . . . . . . . . . 16 (𝑦𝑥) = (𝑥𝑦)
5452, 53syl6eq 2660 . . . . . . . . . . . . . . 15 (𝑤 = 𝑦 → (𝑤𝑥) = (𝑥𝑦))
5554sseq2d 3596 . . . . . . . . . . . . . 14 (𝑤 = 𝑦 → (𝑧 ⊆ (𝑤𝑥) ↔ 𝑧 ⊆ (𝑥𝑦)))
5655rexbidv 3034 . . . . . . . . . . . . 13 (𝑤 = 𝑦 → (∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑤𝑥) ↔ ∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)))
5751, 56cbvral2v 3155 . . . . . . . . . . . 12 (∀𝑦𝐺𝑤𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑤𝑦) ↔ ∀𝑥𝐺𝑦𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
5843, 48, 573bitri 285 . . . . . . . . . . 11 (∀𝑥𝐹𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ↔ ∀𝑥𝐺𝑦𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
5958biimpi 205 . . . . . . . . . 10 (∀𝑥𝐹𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) → ∀𝑥𝐺𝑦𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
60 ssun2 3739 . . . . . . . . . . . . . 14 𝐺 ⊆ (𝐹𝐺)
61 fbasssin 21450 . . . . . . . . . . . . . 14 ((𝐺 ∈ (fBas‘𝑋) ∧ 𝑥𝐺𝑦𝐺) → ∃𝑧𝐺 𝑧 ⊆ (𝑥𝑦))
62 ssrexv 3630 . . . . . . . . . . . . . 14 (𝐺 ⊆ (𝐹𝐺) → (∃𝑧𝐺 𝑧 ⊆ (𝑥𝑦) → ∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)))
6360, 61, 62mpsyl 66 . . . . . . . . . . . . 13 ((𝐺 ∈ (fBas‘𝑋) ∧ 𝑥𝐺𝑦𝐺) → ∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
64633expb 1258 . . . . . . . . . . . 12 ((𝐺 ∈ (fBas‘𝑋) ∧ (𝑥𝐺𝑦𝐺)) → ∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
6564ralrimivva 2954 . . . . . . . . . . 11 (𝐺 ∈ (fBas‘𝑋) → ∀𝑥𝐺𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
6665adantl 481 . . . . . . . . . 10 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → ∀𝑥𝐺𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
6759, 66anim12i 588 . . . . . . . . 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 3046 . . . . . . . . 9 (∀𝑥𝐺 (∀𝑦𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ∧ ∀𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)) ↔ (∀𝑥𝐺𝑦𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ∧ ∀𝑥𝐺𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)))
7039ralimi 2936 . . . . . . . . 9 (∀𝑥𝐺 (∀𝑦𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ∧ ∀𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)) → ∀𝑥𝐺𝑦 ∈ (𝐹𝐺)∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
7169, 70sylbir 224 . . . . . . . 8 ((∀𝑥𝐺𝑦𝐹𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ∧ ∀𝑥𝐺𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)) → ∀𝑥𝐺𝑦 ∈ (𝐹𝐺)∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
7268, 71syl6 34 . . . . . . 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 3757 . . . . . 6 ((∀𝑥𝐹𝑦 ∈ (𝐹𝐺)∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) ∧ ∀𝑥𝐺𝑦 ∈ (𝐹𝐺)∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)) → ∀𝑥 ∈ (𝐹𝐺)∀𝑦 ∈ (𝐹𝐺)∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦))
7573, 74syl6 34 . . . . 5 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (∀𝑥𝐹𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) → ∀𝑥 ∈ (𝐹𝐺)∀𝑦 ∈ (𝐹𝐺)∃𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦)))
7619, 29, 753jcad 1236 . . . 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 6130 . . . . 5 (𝐹 ∈ (fBas‘𝑋) → 𝑋 ∈ dom fBas)
7978adantr 480 . . . 4 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → 𝑋 ∈ dom fBas)
80 isfbas2 21449 . . . 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 248 . 2 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (∀𝑥𝐹𝑦𝐺𝑧 ∈ (𝐹𝐺)𝑧 ⊆ (𝑥𝑦) → (𝐹𝐺) ∈ (fBas‘𝑋)))
837, 82impbid2 215 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 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   ∈ wcel 1977   ≠ wne 2780   ∉ wnel 2781  ∀wral 2896  ∃wrex 2897   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  dom cdm 5038  ‘cfv 5804  fBascfbas 19555 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fv 5812  df-fbas 19564 This theorem is referenced by: (None)
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