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Theorem tgcmp 22896
Description: A topology generated by a basis is compact iff open covers drawn from the basis have finite subcovers. (See also alexsub 23540, which further specializes to subbases, assuming the ultrafilter lemma.) (Contributed by Mario Carneiro, 26-Aug-2015.)
Assertion
Ref Expression
tgcmp ((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) → ((topGen‘𝐵) ∈ Comp ↔ ∀𝑦 ∈ 𝒫 𝐵(𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧)))
Distinct variable groups:   𝑦,𝑧,𝐵   𝑦,𝑋,𝑧

Proof of Theorem tgcmp
Dummy variables 𝑡 𝑓 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2732 . . . . 5 (topGen‘𝐵) = (topGen‘𝐵)
21iscmp 22883 . . . 4 ((topGen‘𝐵) ∈ Comp ↔ ((topGen‘𝐵) ∈ Top ∧ ∀𝑦 ∈ 𝒫 (topGen‘𝐵)( (topGen‘𝐵) = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) (topGen‘𝐵) = 𝑧)))
32simprbi 497 . . 3 ((topGen‘𝐵) ∈ Comp → ∀𝑦 ∈ 𝒫 (topGen‘𝐵)( (topGen‘𝐵) = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) (topGen‘𝐵) = 𝑧))
4 unitg 22461 . . . . . . . 8 (𝐵 ∈ TopBases → (topGen‘𝐵) = 𝐵)
5 eqtr3 2758 . . . . . . . 8 (( (topGen‘𝐵) = 𝐵𝑋 = 𝐵) → (topGen‘𝐵) = 𝑋)
64, 5sylan 580 . . . . . . 7 ((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) → (topGen‘𝐵) = 𝑋)
76eqeq1d 2734 . . . . . 6 ((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) → ( (topGen‘𝐵) = 𝑦𝑋 = 𝑦))
86eqeq1d 2734 . . . . . . 7 ((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) → ( (topGen‘𝐵) = 𝑧𝑋 = 𝑧))
98rexbidv 3178 . . . . . 6 ((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) → (∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) (topGen‘𝐵) = 𝑧 ↔ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧))
107, 9imbi12d 344 . . . . 5 ((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) → (( (topGen‘𝐵) = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) (topGen‘𝐵) = 𝑧) ↔ (𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧)))
1110ralbidv 3177 . . . 4 ((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) → (∀𝑦 ∈ 𝒫 (topGen‘𝐵)( (topGen‘𝐵) = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) (topGen‘𝐵) = 𝑧) ↔ ∀𝑦 ∈ 𝒫 (topGen‘𝐵)(𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧)))
12 bastg 22460 . . . . . . 7 (𝐵 ∈ TopBases → 𝐵 ⊆ (topGen‘𝐵))
1312adantr 481 . . . . . 6 ((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) → 𝐵 ⊆ (topGen‘𝐵))
1413sspwd 4614 . . . . 5 ((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) → 𝒫 𝐵 ⊆ 𝒫 (topGen‘𝐵))
15 ssralv 4049 . . . . 5 (𝒫 𝐵 ⊆ 𝒫 (topGen‘𝐵) → (∀𝑦 ∈ 𝒫 (topGen‘𝐵)(𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧) → ∀𝑦 ∈ 𝒫 𝐵(𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧)))
1614, 15syl 17 . . . 4 ((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) → (∀𝑦 ∈ 𝒫 (topGen‘𝐵)(𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧) → ∀𝑦 ∈ 𝒫 𝐵(𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧)))
1711, 16sylbid 239 . . 3 ((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) → (∀𝑦 ∈ 𝒫 (topGen‘𝐵)( (topGen‘𝐵) = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) (topGen‘𝐵) = 𝑧) → ∀𝑦 ∈ 𝒫 𝐵(𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧)))
183, 17syl5 34 . 2 ((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) → ((topGen‘𝐵) ∈ Comp → ∀𝑦 ∈ 𝒫 𝐵(𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧)))
19 elpwi 4608 . . . . 5 (𝑢 ∈ 𝒫 (topGen‘𝐵) → 𝑢 ⊆ (topGen‘𝐵))
20 simprr 771 . . . . . . . . . . 11 (((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) → 𝑋 = 𝑢)
21 simprl 769 . . . . . . . . . . . . . . . . . 18 (((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) → 𝑢 ⊆ (topGen‘𝐵))
2221sselda 3981 . . . . . . . . . . . . . . . . 17 ((((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) ∧ 𝑡𝑢) → 𝑡 ∈ (topGen‘𝐵))
2322adantrr 715 . . . . . . . . . . . . . . . 16 ((((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) ∧ (𝑡𝑢𝑦𝑡)) → 𝑡 ∈ (topGen‘𝐵))
24 simprr 771 . . . . . . . . . . . . . . . 16 ((((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) ∧ (𝑡𝑢𝑦𝑡)) → 𝑦𝑡)
25 tg2 22459 . . . . . . . . . . . . . . . 16 ((𝑡 ∈ (topGen‘𝐵) ∧ 𝑦𝑡) → ∃𝑤𝐵 (𝑦𝑤𝑤𝑡))
2623, 24, 25syl2anc 584 . . . . . . . . . . . . . . 15 ((((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) ∧ (𝑡𝑢𝑦𝑡)) → ∃𝑤𝐵 (𝑦𝑤𝑤𝑡))
2726expr 457 . . . . . . . . . . . . . 14 ((((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) ∧ 𝑡𝑢) → (𝑦𝑡 → ∃𝑤𝐵 (𝑦𝑤𝑤𝑡)))
2827reximdva 3168 . . . . . . . . . . . . 13 (((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) → (∃𝑡𝑢 𝑦𝑡 → ∃𝑡𝑢𝑤𝐵 (𝑦𝑤𝑤𝑡)))
29 eluni2 4911 . . . . . . . . . . . . 13 (𝑦 𝑢 ↔ ∃𝑡𝑢 𝑦𝑡)
30 elunirab 4923 . . . . . . . . . . . . . 14 (𝑦 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ↔ ∃𝑤𝐵 (𝑦𝑤 ∧ ∃𝑡𝑢 𝑤𝑡))
31 r19.42v 3190 . . . . . . . . . . . . . . 15 (∃𝑡𝑢 (𝑦𝑤𝑤𝑡) ↔ (𝑦𝑤 ∧ ∃𝑡𝑢 𝑤𝑡))
3231rexbii 3094 . . . . . . . . . . . . . 14 (∃𝑤𝐵𝑡𝑢 (𝑦𝑤𝑤𝑡) ↔ ∃𝑤𝐵 (𝑦𝑤 ∧ ∃𝑡𝑢 𝑤𝑡))
33 rexcom 3287 . . . . . . . . . . . . . 14 (∃𝑤𝐵𝑡𝑢 (𝑦𝑤𝑤𝑡) ↔ ∃𝑡𝑢𝑤𝐵 (𝑦𝑤𝑤𝑡))
3430, 32, 333bitr2i 298 . . . . . . . . . . . . 13 (𝑦 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ↔ ∃𝑡𝑢𝑤𝐵 (𝑦𝑤𝑤𝑡))
3528, 29, 343imtr4g 295 . . . . . . . . . . . 12 (((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) → (𝑦 𝑢𝑦 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡}))
3635ssrdv 3987 . . . . . . . . . . 11 (((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) → 𝑢 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡})
3720, 36eqsstrd 4019 . . . . . . . . . 10 (((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) → 𝑋 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡})
38 ssrab2 4076 . . . . . . . . . . . 12 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ⊆ 𝐵
3938unissi 4916 . . . . . . . . . . 11 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ⊆ 𝐵
40 simplr 767 . . . . . . . . . . 11 (((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) → 𝑋 = 𝐵)
4139, 40sseqtrrid 4034 . . . . . . . . . 10 (((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) → {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ⊆ 𝑋)
4237, 41eqssd 3998 . . . . . . . . 9 (((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) → 𝑋 = {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡})
43 elpw2g 5343 . . . . . . . . . . . 12 (𝐵 ∈ TopBases → ({𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∈ 𝒫 𝐵 ↔ {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ⊆ 𝐵))
4443ad2antrr 724 . . . . . . . . . . 11 (((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) → ({𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∈ 𝒫 𝐵 ↔ {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ⊆ 𝐵))
4538, 44mpbiri 257 . . . . . . . . . 10 (((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) → {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∈ 𝒫 𝐵)
46 unieq 4918 . . . . . . . . . . . . 13 (𝑦 = {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} → 𝑦 = {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡})
4746eqeq2d 2743 . . . . . . . . . . . 12 (𝑦 = {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} → (𝑋 = 𝑦𝑋 = {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡}))
48 pweq 4615 . . . . . . . . . . . . . 14 (𝑦 = {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} → 𝒫 𝑦 = 𝒫 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡})
4948ineq1d 4210 . . . . . . . . . . . . 13 (𝑦 = {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} → (𝒫 𝑦 ∩ Fin) = (𝒫 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∩ Fin))
5049rexeqdv 3326 . . . . . . . . . . . 12 (𝑦 = {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} → (∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧 ↔ ∃𝑧 ∈ (𝒫 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∩ Fin)𝑋 = 𝑧))
5147, 50imbi12d 344 . . . . . . . . . . 11 (𝑦 = {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} → ((𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧) ↔ (𝑋 = {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} → ∃𝑧 ∈ (𝒫 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∩ Fin)𝑋 = 𝑧)))
5251rspcv 3608 . . . . . . . . . 10 ({𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∈ 𝒫 𝐵 → (∀𝑦 ∈ 𝒫 𝐵(𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧) → (𝑋 = {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} → ∃𝑧 ∈ (𝒫 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∩ Fin)𝑋 = 𝑧)))
5345, 52syl 17 . . . . . . . . 9 (((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) → (∀𝑦 ∈ 𝒫 𝐵(𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧) → (𝑋 = {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} → ∃𝑧 ∈ (𝒫 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∩ Fin)𝑋 = 𝑧)))
5442, 53mpid 44 . . . . . . . 8 (((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) → (∀𝑦 ∈ 𝒫 𝐵(𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧) → ∃𝑧 ∈ (𝒫 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∩ Fin)𝑋 = 𝑧))
55 elfpw 9350 . . . . . . . . . . . . 13 (𝑧 ∈ (𝒫 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∩ Fin) ↔ (𝑧 ⊆ {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∧ 𝑧 ∈ Fin))
5655simprbi 497 . . . . . . . . . . . 12 (𝑧 ∈ (𝒫 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∩ Fin) → 𝑧 ∈ Fin)
5756ad2antrl 726 . . . . . . . . . . 11 ((((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) ∧ (𝑧 ∈ (𝒫 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∩ Fin) ∧ 𝑋 = 𝑧)) → 𝑧 ∈ Fin)
5855simplbi 498 . . . . . . . . . . . . 13 (𝑧 ∈ (𝒫 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∩ Fin) → 𝑧 ⊆ {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡})
5958ad2antrl 726 . . . . . . . . . . . 12 ((((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) ∧ (𝑧 ∈ (𝒫 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∩ Fin) ∧ 𝑋 = 𝑧)) → 𝑧 ⊆ {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡})
60 ssrab 4069 . . . . . . . . . . . . 13 (𝑧 ⊆ {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ↔ (𝑧𝐵 ∧ ∀𝑤𝑧𝑡𝑢 𝑤𝑡))
6160simprbi 497 . . . . . . . . . . . 12 (𝑧 ⊆ {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} → ∀𝑤𝑧𝑡𝑢 𝑤𝑡)
6259, 61syl 17 . . . . . . . . . . 11 ((((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) ∧ (𝑧 ∈ (𝒫 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∩ Fin) ∧ 𝑋 = 𝑧)) → ∀𝑤𝑧𝑡𝑢 𝑤𝑡)
63 sseq2 4007 . . . . . . . . . . . 12 (𝑡 = (𝑓𝑤) → (𝑤𝑡𝑤 ⊆ (𝑓𝑤)))
6463ac6sfi 9283 . . . . . . . . . . 11 ((𝑧 ∈ Fin ∧ ∀𝑤𝑧𝑡𝑢 𝑤𝑡) → ∃𝑓(𝑓:𝑧𝑢 ∧ ∀𝑤𝑧 𝑤 ⊆ (𝑓𝑤)))
6557, 62, 64syl2anc 584 . . . . . . . . . 10 ((((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) ∧ (𝑧 ∈ (𝒫 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∩ Fin) ∧ 𝑋 = 𝑧)) → ∃𝑓(𝑓:𝑧𝑢 ∧ ∀𝑤𝑧 𝑤 ⊆ (𝑓𝑤)))
66 frn 6721 . . . . . . . . . . . . 13 (𝑓:𝑧𝑢 → ran 𝑓𝑢)
6766ad2antrl 726 . . . . . . . . . . . 12 (((((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) ∧ (𝑧 ∈ (𝒫 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∩ Fin) ∧ 𝑋 = 𝑧)) ∧ (𝑓:𝑧𝑢 ∧ ∀𝑤𝑧 𝑤 ⊆ (𝑓𝑤))) → ran 𝑓𝑢)
68 ffn 6714 . . . . . . . . . . . . . . 15 (𝑓:𝑧𝑢𝑓 Fn 𝑧)
69 dffn4 6808 . . . . . . . . . . . . . . 15 (𝑓 Fn 𝑧𝑓:𝑧onto→ran 𝑓)
7068, 69sylib 217 . . . . . . . . . . . . . 14 (𝑓:𝑧𝑢𝑓:𝑧onto→ran 𝑓)
7170adantr 481 . . . . . . . . . . . . 13 ((𝑓:𝑧𝑢 ∧ ∀𝑤𝑧 𝑤 ⊆ (𝑓𝑤)) → 𝑓:𝑧onto→ran 𝑓)
72 fofi 9334 . . . . . . . . . . . . 13 ((𝑧 ∈ Fin ∧ 𝑓:𝑧onto→ran 𝑓) → ran 𝑓 ∈ Fin)
7357, 71, 72syl2an 596 . . . . . . . . . . . 12 (((((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) ∧ (𝑧 ∈ (𝒫 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∩ Fin) ∧ 𝑋 = 𝑧)) ∧ (𝑓:𝑧𝑢 ∧ ∀𝑤𝑧 𝑤 ⊆ (𝑓𝑤))) → ran 𝑓 ∈ Fin)
74 elfpw 9350 . . . . . . . . . . . 12 (ran 𝑓 ∈ (𝒫 𝑢 ∩ Fin) ↔ (ran 𝑓𝑢 ∧ ran 𝑓 ∈ Fin))
7567, 73, 74sylanbrc 583 . . . . . . . . . . 11 (((((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) ∧ (𝑧 ∈ (𝒫 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∩ Fin) ∧ 𝑋 = 𝑧)) ∧ (𝑓:𝑧𝑢 ∧ ∀𝑤𝑧 𝑤 ⊆ (𝑓𝑤))) → ran 𝑓 ∈ (𝒫 𝑢 ∩ Fin))
76 simplrr 776 . . . . . . . . . . . . 13 (((((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) ∧ (𝑧 ∈ (𝒫 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∩ Fin) ∧ 𝑋 = 𝑧)) ∧ (𝑓:𝑧𝑢 ∧ ∀𝑤𝑧 𝑤 ⊆ (𝑓𝑤))) → 𝑋 = 𝑧)
77 uniiun 5060 . . . . . . . . . . . . . . . 16 𝑧 = 𝑤𝑧 𝑤
78 ss2iun 5014 . . . . . . . . . . . . . . . 16 (∀𝑤𝑧 𝑤 ⊆ (𝑓𝑤) → 𝑤𝑧 𝑤 𝑤𝑧 (𝑓𝑤))
7977, 78eqsstrid 4029 . . . . . . . . . . . . . . 15 (∀𝑤𝑧 𝑤 ⊆ (𝑓𝑤) → 𝑧 𝑤𝑧 (𝑓𝑤))
8079ad2antll 727 . . . . . . . . . . . . . 14 (((((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) ∧ (𝑧 ∈ (𝒫 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∩ Fin) ∧ 𝑋 = 𝑧)) ∧ (𝑓:𝑧𝑢 ∧ ∀𝑤𝑧 𝑤 ⊆ (𝑓𝑤))) → 𝑧 𝑤𝑧 (𝑓𝑤))
81 fniunfv 7242 . . . . . . . . . . . . . . . 16 (𝑓 Fn 𝑧 𝑤𝑧 (𝑓𝑤) = ran 𝑓)
8268, 81syl 17 . . . . . . . . . . . . . . 15 (𝑓:𝑧𝑢 𝑤𝑧 (𝑓𝑤) = ran 𝑓)
8382ad2antrl 726 . . . . . . . . . . . . . 14 (((((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) ∧ (𝑧 ∈ (𝒫 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∩ Fin) ∧ 𝑋 = 𝑧)) ∧ (𝑓:𝑧𝑢 ∧ ∀𝑤𝑧 𝑤 ⊆ (𝑓𝑤))) → 𝑤𝑧 (𝑓𝑤) = ran 𝑓)
8480, 83sseqtrd 4021 . . . . . . . . . . . . 13 (((((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) ∧ (𝑧 ∈ (𝒫 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∩ Fin) ∧ 𝑋 = 𝑧)) ∧ (𝑓:𝑧𝑢 ∧ ∀𝑤𝑧 𝑤 ⊆ (𝑓𝑤))) → 𝑧 ran 𝑓)
8576, 84eqsstrd 4019 . . . . . . . . . . . 12 (((((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) ∧ (𝑧 ∈ (𝒫 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∩ Fin) ∧ 𝑋 = 𝑧)) ∧ (𝑓:𝑧𝑢 ∧ ∀𝑤𝑧 𝑤 ⊆ (𝑓𝑤))) → 𝑋 ran 𝑓)
8667unissd 4917 . . . . . . . . . . . . 13 (((((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) ∧ (𝑧 ∈ (𝒫 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∩ Fin) ∧ 𝑋 = 𝑧)) ∧ (𝑓:𝑧𝑢 ∧ ∀𝑤𝑧 𝑤 ⊆ (𝑓𝑤))) → ran 𝑓 𝑢)
8720ad2antrr 724 . . . . . . . . . . . . 13 (((((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) ∧ (𝑧 ∈ (𝒫 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∩ Fin) ∧ 𝑋 = 𝑧)) ∧ (𝑓:𝑧𝑢 ∧ ∀𝑤𝑧 𝑤 ⊆ (𝑓𝑤))) → 𝑋 = 𝑢)
8886, 87sseqtrrd 4022 . . . . . . . . . . . 12 (((((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) ∧ (𝑧 ∈ (𝒫 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∩ Fin) ∧ 𝑋 = 𝑧)) ∧ (𝑓:𝑧𝑢 ∧ ∀𝑤𝑧 𝑤 ⊆ (𝑓𝑤))) → ran 𝑓𝑋)
8985, 88eqssd 3998 . . . . . . . . . . 11 (((((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) ∧ (𝑧 ∈ (𝒫 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∩ Fin) ∧ 𝑋 = 𝑧)) ∧ (𝑓:𝑧𝑢 ∧ ∀𝑤𝑧 𝑤 ⊆ (𝑓𝑤))) → 𝑋 = ran 𝑓)
90 unieq 4918 . . . . . . . . . . . 12 (𝑣 = ran 𝑓 𝑣 = ran 𝑓)
9190rspceeqv 3632 . . . . . . . . . . 11 ((ran 𝑓 ∈ (𝒫 𝑢 ∩ Fin) ∧ 𝑋 = ran 𝑓) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = 𝑣)
9275, 89, 91syl2anc 584 . . . . . . . . . 10 (((((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) ∧ (𝑧 ∈ (𝒫 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∩ Fin) ∧ 𝑋 = 𝑧)) ∧ (𝑓:𝑧𝑢 ∧ ∀𝑤𝑧 𝑤 ⊆ (𝑓𝑤))) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = 𝑣)
9365, 92exlimddv 1938 . . . . . . . . 9 ((((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) ∧ (𝑧 ∈ (𝒫 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∩ Fin) ∧ 𝑋 = 𝑧)) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = 𝑣)
9493rexlimdvaa 3156 . . . . . . . 8 (((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) → (∃𝑧 ∈ (𝒫 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∩ Fin)𝑋 = 𝑧 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = 𝑣))
9554, 94syld 47 . . . . . . 7 (((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) → (∀𝑦 ∈ 𝒫 𝐵(𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = 𝑣))
9695expr 457 . . . . . 6 (((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ 𝑢 ⊆ (topGen‘𝐵)) → (𝑋 = 𝑢 → (∀𝑦 ∈ 𝒫 𝐵(𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = 𝑣)))
9796com23 86 . . . . 5 (((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ 𝑢 ⊆ (topGen‘𝐵)) → (∀𝑦 ∈ 𝒫 𝐵(𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧) → (𝑋 = 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = 𝑣)))
9819, 97sylan2 593 . . . 4 (((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ 𝑢 ∈ 𝒫 (topGen‘𝐵)) → (∀𝑦 ∈ 𝒫 𝐵(𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧) → (𝑋 = 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = 𝑣)))
9998ralrimdva 3154 . . 3 ((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) → (∀𝑦 ∈ 𝒫 𝐵(𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧) → ∀𝑢 ∈ 𝒫 (topGen‘𝐵)(𝑋 = 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = 𝑣)))
100 tgcl 22463 . . . . . 6 (𝐵 ∈ TopBases → (topGen‘𝐵) ∈ Top)
101100adantr 481 . . . . 5 ((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) → (topGen‘𝐵) ∈ Top)
1021iscmp 22883 . . . . . 6 ((topGen‘𝐵) ∈ Comp ↔ ((topGen‘𝐵) ∈ Top ∧ ∀𝑢 ∈ 𝒫 (topGen‘𝐵)( (topGen‘𝐵) = 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin) (topGen‘𝐵) = 𝑣)))
103102baib 536 . . . . 5 ((topGen‘𝐵) ∈ Top → ((topGen‘𝐵) ∈ Comp ↔ ∀𝑢 ∈ 𝒫 (topGen‘𝐵)( (topGen‘𝐵) = 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin) (topGen‘𝐵) = 𝑣)))
104101, 103syl 17 . . . 4 ((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) → ((topGen‘𝐵) ∈ Comp ↔ ∀𝑢 ∈ 𝒫 (topGen‘𝐵)( (topGen‘𝐵) = 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin) (topGen‘𝐵) = 𝑣)))
1056eqeq1d 2734 . . . . . 6 ((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) → ( (topGen‘𝐵) = 𝑢𝑋 = 𝑢))
1066eqeq1d 2734 . . . . . . 7 ((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) → ( (topGen‘𝐵) = 𝑣𝑋 = 𝑣))
107106rexbidv 3178 . . . . . 6 ((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) → (∃𝑣 ∈ (𝒫 𝑢 ∩ Fin) (topGen‘𝐵) = 𝑣 ↔ ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = 𝑣))
108105, 107imbi12d 344 . . . . 5 ((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) → (( (topGen‘𝐵) = 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin) (topGen‘𝐵) = 𝑣) ↔ (𝑋 = 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = 𝑣)))
109108ralbidv 3177 . . . 4 ((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) → (∀𝑢 ∈ 𝒫 (topGen‘𝐵)( (topGen‘𝐵) = 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin) (topGen‘𝐵) = 𝑣) ↔ ∀𝑢 ∈ 𝒫 (topGen‘𝐵)(𝑋 = 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = 𝑣)))
110104, 109bitrd 278 . . 3 ((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) → ((topGen‘𝐵) ∈ Comp ↔ ∀𝑢 ∈ 𝒫 (topGen‘𝐵)(𝑋 = 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = 𝑣)))
11199, 110sylibrd 258 . 2 ((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) → (∀𝑦 ∈ 𝒫 𝐵(𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧) → (topGen‘𝐵) ∈ Comp))
11218, 111impbid 211 1 ((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) → ((topGen‘𝐵) ∈ Comp ↔ ∀𝑦 ∈ 𝒫 𝐵(𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wex 1781  wcel 2106  wral 3061  wrex 3070  {crab 3432  cin 3946  wss 3947  𝒫 cpw 4601   cuni 4907   ciun 4996  ran crn 5676   Fn wfn 6535  wf 6536  ontowfo 6538  cfv 6540  Fincfn 8935  topGenctg 17379  Topctop 22386  TopBasesctb 22439  Compccmp 22881
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-om 7852  df-1o 8462  df-er 8699  df-en 8936  df-dom 8937  df-fin 8939  df-topgen 17385  df-top 22387  df-bases 22440  df-cmp 22882
This theorem is referenced by: (None)
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