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Theorem tgcmp 21406
Description: A topology generated by a basis is compact iff open covers drawn from the basis have finite subcovers. (See also alexsub 22050, 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 2760 . . . . 5 (topGen‘𝐵) = (topGen‘𝐵)
21iscmp 21393 . . . 4 ((topGen‘𝐵) ∈ Comp ↔ ((topGen‘𝐵) ∈ Top ∧ ∀𝑦 ∈ 𝒫 (topGen‘𝐵)( (topGen‘𝐵) = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) (topGen‘𝐵) = 𝑧)))
32simprbi 483 . . 3 ((topGen‘𝐵) ∈ Comp → ∀𝑦 ∈ 𝒫 (topGen‘𝐵)( (topGen‘𝐵) = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) (topGen‘𝐵) = 𝑧))
4 unitg 20973 . . . . . . . 8 (𝐵 ∈ TopBases → (topGen‘𝐵) = 𝐵)
5 eqtr3 2781 . . . . . . . 8 (( (topGen‘𝐵) = 𝐵𝑋 = 𝐵) → (topGen‘𝐵) = 𝑋)
64, 5sylan 489 . . . . . . 7 ((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) → (topGen‘𝐵) = 𝑋)
76eqeq1d 2762 . . . . . 6 ((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) → ( (topGen‘𝐵) = 𝑦𝑋 = 𝑦))
86eqeq1d 2762 . . . . . . 7 ((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) → ( (topGen‘𝐵) = 𝑧𝑋 = 𝑧))
98rexbidv 3190 . . . . . 6 ((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) → (∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) (topGen‘𝐵) = 𝑧 ↔ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧))
107, 9imbi12d 333 . . . . 5 ((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) → (( (topGen‘𝐵) = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) (topGen‘𝐵) = 𝑧) ↔ (𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧)))
1110ralbidv 3124 . . . 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 20972 . . . . . . 7 (𝐵 ∈ TopBases → 𝐵 ⊆ (topGen‘𝐵))
1312adantr 472 . . . . . 6 ((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) → 𝐵 ⊆ (topGen‘𝐵))
14 sspwb 5066 . . . . . 6 (𝐵 ⊆ (topGen‘𝐵) ↔ 𝒫 𝐵 ⊆ 𝒫 (topGen‘𝐵))
1513, 14sylib 208 . . . . 5 ((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) → 𝒫 𝐵 ⊆ 𝒫 (topGen‘𝐵))
16 ssralv 3807 . . . . 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)))
1715, 16syl 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)))
1811, 17sylbid 230 . . 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)))
193, 18syl5 34 . 2 ((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) → ((topGen‘𝐵) ∈ Comp → ∀𝑦 ∈ 𝒫 𝐵(𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧)))
20 elpwi 4312 . . . . 5 (𝑢 ∈ 𝒫 (topGen‘𝐵) → 𝑢 ⊆ (topGen‘𝐵))
21 simprr 813 . . . . . . . . . . 11 (((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) → 𝑋 = 𝑢)
22 simprl 811 . . . . . . . . . . . . . . . . . 18 (((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) → 𝑢 ⊆ (topGen‘𝐵))
2322sselda 3744 . . . . . . . . . . . . . . . . 17 ((((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) ∧ 𝑡𝑢) → 𝑡 ∈ (topGen‘𝐵))
2423adantrr 755 . . . . . . . . . . . . . . . 16 ((((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) ∧ (𝑡𝑢𝑦𝑡)) → 𝑡 ∈ (topGen‘𝐵))
25 simprr 813 . . . . . . . . . . . . . . . 16 ((((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) ∧ (𝑡𝑢𝑦𝑡)) → 𝑦𝑡)
26 tg2 20971 . . . . . . . . . . . . . . . 16 ((𝑡 ∈ (topGen‘𝐵) ∧ 𝑦𝑡) → ∃𝑤𝐵 (𝑦𝑤𝑤𝑡))
2724, 25, 26syl2anc 696 . . . . . . . . . . . . . . 15 ((((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) ∧ (𝑡𝑢𝑦𝑡)) → ∃𝑤𝐵 (𝑦𝑤𝑤𝑡))
2827expr 644 . . . . . . . . . . . . . 14 ((((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) ∧ 𝑡𝑢) → (𝑦𝑡 → ∃𝑤𝐵 (𝑦𝑤𝑤𝑡)))
2928reximdva 3155 . . . . . . . . . . . . 13 (((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) → (∃𝑡𝑢 𝑦𝑡 → ∃𝑡𝑢𝑤𝐵 (𝑦𝑤𝑤𝑡)))
30 eluni2 4592 . . . . . . . . . . . . 13 (𝑦 𝑢 ↔ ∃𝑡𝑢 𝑦𝑡)
31 elunirab 4600 . . . . . . . . . . . . . 14 (𝑦 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ↔ ∃𝑤𝐵 (𝑦𝑤 ∧ ∃𝑡𝑢 𝑤𝑡))
32 r19.42v 3230 . . . . . . . . . . . . . . 15 (∃𝑡𝑢 (𝑦𝑤𝑤𝑡) ↔ (𝑦𝑤 ∧ ∃𝑡𝑢 𝑤𝑡))
3332rexbii 3179 . . . . . . . . . . . . . 14 (∃𝑤𝐵𝑡𝑢 (𝑦𝑤𝑤𝑡) ↔ ∃𝑤𝐵 (𝑦𝑤 ∧ ∃𝑡𝑢 𝑤𝑡))
34 rexcom 3237 . . . . . . . . . . . . . 14 (∃𝑤𝐵𝑡𝑢 (𝑦𝑤𝑤𝑡) ↔ ∃𝑡𝑢𝑤𝐵 (𝑦𝑤𝑤𝑡))
3531, 33, 343bitr2i 288 . . . . . . . . . . . . 13 (𝑦 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ↔ ∃𝑡𝑢𝑤𝐵 (𝑦𝑤𝑤𝑡))
3629, 30, 353imtr4g 285 . . . . . . . . . . . 12 (((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) → (𝑦 𝑢𝑦 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡}))
3736ssrdv 3750 . . . . . . . . . . 11 (((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) → 𝑢 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡})
3821, 37eqsstrd 3780 . . . . . . . . . 10 (((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) → 𝑋 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡})
39 ssrab2 3828 . . . . . . . . . . . 12 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ⊆ 𝐵
4039unissi 4613 . . . . . . . . . . 11 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ⊆ 𝐵
41 simplr 809 . . . . . . . . . . 11 (((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) → 𝑋 = 𝐵)
4240, 41syl5sseqr 3795 . . . . . . . . . 10 (((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) → {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ⊆ 𝑋)
4338, 42eqssd 3761 . . . . . . . . 9 (((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) → 𝑋 = {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡})
44 elpw2g 4976 . . . . . . . . . . . 12 (𝐵 ∈ TopBases → ({𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∈ 𝒫 𝐵 ↔ {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ⊆ 𝐵))
4544ad2antrr 764 . . . . . . . . . . 11 (((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) → ({𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∈ 𝒫 𝐵 ↔ {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ⊆ 𝐵))
4639, 45mpbiri 248 . . . . . . . . . 10 (((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) → {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∈ 𝒫 𝐵)
47 unieq 4596 . . . . . . . . . . . . 13 (𝑦 = {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} → 𝑦 = {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡})
4847eqeq2d 2770 . . . . . . . . . . . 12 (𝑦 = {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} → (𝑋 = 𝑦𝑋 = {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡}))
49 pweq 4305 . . . . . . . . . . . . . 14 (𝑦 = {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} → 𝒫 𝑦 = 𝒫 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡})
5049ineq1d 3956 . . . . . . . . . . . . 13 (𝑦 = {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} → (𝒫 𝑦 ∩ Fin) = (𝒫 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∩ Fin))
5150rexeqdv 3284 . . . . . . . . . . . 12 (𝑦 = {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} → (∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧 ↔ ∃𝑧 ∈ (𝒫 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∩ Fin)𝑋 = 𝑧))
5248, 51imbi12d 333 . . . . . . . . . . 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)))
5352rspcv 3445 . . . . . . . . . 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)))
5446, 53syl 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)))
5543, 54mpid 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))
56 elfpw 8433 . . . . . . . . . . . . 13 (𝑧 ∈ (𝒫 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∩ Fin) ↔ (𝑧 ⊆ {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∧ 𝑧 ∈ Fin))
5756simprbi 483 . . . . . . . . . . . 12 (𝑧 ∈ (𝒫 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∩ Fin) → 𝑧 ∈ Fin)
5857ad2antrl 766 . . . . . . . . . . 11 ((((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) ∧ (𝑧 ∈ (𝒫 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∩ Fin) ∧ 𝑋 = 𝑧)) → 𝑧 ∈ Fin)
5956simplbi 478 . . . . . . . . . . . . 13 (𝑧 ∈ (𝒫 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∩ Fin) → 𝑧 ⊆ {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡})
6059ad2antrl 766 . . . . . . . . . . . 12 ((((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) ∧ (𝑧 ∈ (𝒫 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∩ Fin) ∧ 𝑋 = 𝑧)) → 𝑧 ⊆ {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡})
61 ssrab 3821 . . . . . . . . . . . . 13 (𝑧 ⊆ {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ↔ (𝑧𝐵 ∧ ∀𝑤𝑧𝑡𝑢 𝑤𝑡))
6261simprbi 483 . . . . . . . . . . . 12 (𝑧 ⊆ {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} → ∀𝑤𝑧𝑡𝑢 𝑤𝑡)
6360, 62syl 17 . . . . . . . . . . 11 ((((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) ∧ (𝑧 ∈ (𝒫 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∩ Fin) ∧ 𝑋 = 𝑧)) → ∀𝑤𝑧𝑡𝑢 𝑤𝑡)
64 sseq2 3768 . . . . . . . . . . . 12 (𝑡 = (𝑓𝑤) → (𝑤𝑡𝑤 ⊆ (𝑓𝑤)))
6564ac6sfi 8369 . . . . . . . . . . 11 ((𝑧 ∈ Fin ∧ ∀𝑤𝑧𝑡𝑢 𝑤𝑡) → ∃𝑓(𝑓:𝑧𝑢 ∧ ∀𝑤𝑧 𝑤 ⊆ (𝑓𝑤)))
6658, 63, 65syl2anc 696 . . . . . . . . . 10 ((((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) ∧ (𝑧 ∈ (𝒫 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∩ Fin) ∧ 𝑋 = 𝑧)) → ∃𝑓(𝑓:𝑧𝑢 ∧ ∀𝑤𝑧 𝑤 ⊆ (𝑓𝑤)))
67 frn 6214 . . . . . . . . . . . . 13 (𝑓:𝑧𝑢 → ran 𝑓𝑢)
6867ad2antrl 766 . . . . . . . . . . . 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)
69 ffn 6206 . . . . . . . . . . . . . . 15 (𝑓:𝑧𝑢𝑓 Fn 𝑧)
70 dffn4 6282 . . . . . . . . . . . . . . 15 (𝑓 Fn 𝑧𝑓:𝑧onto→ran 𝑓)
7169, 70sylib 208 . . . . . . . . . . . . . 14 (𝑓:𝑧𝑢𝑓:𝑧onto→ran 𝑓)
7271adantr 472 . . . . . . . . . . . . 13 ((𝑓:𝑧𝑢 ∧ ∀𝑤𝑧 𝑤 ⊆ (𝑓𝑤)) → 𝑓:𝑧onto→ran 𝑓)
73 fofi 8417 . . . . . . . . . . . . 13 ((𝑧 ∈ Fin ∧ 𝑓:𝑧onto→ran 𝑓) → ran 𝑓 ∈ Fin)
7458, 72, 73syl2an 495 . . . . . . . . . . . 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)
75 elfpw 8433 . . . . . . . . . . . 12 (ran 𝑓 ∈ (𝒫 𝑢 ∩ Fin) ↔ (ran 𝑓𝑢 ∧ ran 𝑓 ∈ Fin))
7668, 74, 75sylanbrc 701 . . . . . . . . . . 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))
77 simplrr 820 . . . . . . . . . . . . 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)
78 uniiun 4725 . . . . . . . . . . . . . . . 16 𝑧 = 𝑤𝑧 𝑤
79 ss2iun 4688 . . . . . . . . . . . . . . . 16 (∀𝑤𝑧 𝑤 ⊆ (𝑓𝑤) → 𝑤𝑧 𝑤 𝑤𝑧 (𝑓𝑤))
8078, 79syl5eqss 3790 . . . . . . . . . . . . . . 15 (∀𝑤𝑧 𝑤 ⊆ (𝑓𝑤) → 𝑧 𝑤𝑧 (𝑓𝑤))
8180ad2antll 767 . . . . . . . . . . . . . 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))
82 fniunfv 6668 . . . . . . . . . . . . . . . 16 (𝑓 Fn 𝑧 𝑤𝑧 (𝑓𝑤) = ran 𝑓)
8369, 82syl 17 . . . . . . . . . . . . . . 15 (𝑓:𝑧𝑢 𝑤𝑧 (𝑓𝑤) = ran 𝑓)
8483ad2antrl 766 . . . . . . . . . . . . . 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)
8581, 84sseqtrd 3782 . . . . . . . . . . . . 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)
8677, 85eqsstrd 3780 . . . . . . . . . . . 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)
8768unissd 4614 . . . . . . . . . . . . 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)
8821ad2antrr 764 . . . . . . . . . . . . 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)
8987, 88sseqtr4d 3783 . . . . . . . . . . . 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)
9086, 89eqssd 3761 . . . . . . . . . . 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)
91 unieq 4596 . . . . . . . . . . . . 13 (𝑣 = ran 𝑓 𝑣 = ran 𝑓)
9291eqeq2d 2770 . . . . . . . . . . . 12 (𝑣 = ran 𝑓 → (𝑋 = 𝑣𝑋 = ran 𝑓))
9392rspcev 3449 . . . . . . . . . . 11 ((ran 𝑓 ∈ (𝒫 𝑢 ∩ Fin) ∧ 𝑋 = ran 𝑓) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = 𝑣)
9476, 90, 93syl2anc 696 . . . . . . . . . 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)
9566, 94exlimddv 2012 . . . . . . . . 9 ((((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) ∧ (𝑧 ∈ (𝒫 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∩ Fin) ∧ 𝑋 = 𝑧)) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = 𝑣)
9695rexlimdvaa 3170 . . . . . . . 8 (((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) → (∃𝑧 ∈ (𝒫 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∩ Fin)𝑋 = 𝑧 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = 𝑣))
9755, 96syld 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))
9897expr 644 . . . . . 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)))
9998com23 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)))
10020, 99sylan2 492 . . . 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)))
101100ralrimdva 3107 . . 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)))
102 tgcl 20975 . . . . . 6 (𝐵 ∈ TopBases → (topGen‘𝐵) ∈ Top)
103102adantr 472 . . . . 5 ((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) → (topGen‘𝐵) ∈ Top)
1041iscmp 21393 . . . . . 6 ((topGen‘𝐵) ∈ Comp ↔ ((topGen‘𝐵) ∈ Top ∧ ∀𝑢 ∈ 𝒫 (topGen‘𝐵)( (topGen‘𝐵) = 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin) (topGen‘𝐵) = 𝑣)))
105104baib 982 . . . . 5 ((topGen‘𝐵) ∈ Top → ((topGen‘𝐵) ∈ Comp ↔ ∀𝑢 ∈ 𝒫 (topGen‘𝐵)( (topGen‘𝐵) = 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin) (topGen‘𝐵) = 𝑣)))
106103, 105syl 17 . . . 4 ((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) → ((topGen‘𝐵) ∈ Comp ↔ ∀𝑢 ∈ 𝒫 (topGen‘𝐵)( (topGen‘𝐵) = 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin) (topGen‘𝐵) = 𝑣)))
1076eqeq1d 2762 . . . . . 6 ((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) → ( (topGen‘𝐵) = 𝑢𝑋 = 𝑢))
1086eqeq1d 2762 . . . . . . 7 ((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) → ( (topGen‘𝐵) = 𝑣𝑋 = 𝑣))
109108rexbidv 3190 . . . . . 6 ((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) → (∃𝑣 ∈ (𝒫 𝑢 ∩ Fin) (topGen‘𝐵) = 𝑣 ↔ ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = 𝑣))
110107, 109imbi12d 333 . . . . 5 ((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) → (( (topGen‘𝐵) = 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin) (topGen‘𝐵) = 𝑣) ↔ (𝑋 = 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = 𝑣)))
111110ralbidv 3124 . . . 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)))
112106, 111bitrd 268 . . 3 ((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) → ((topGen‘𝐵) ∈ Comp ↔ ∀𝑢 ∈ 𝒫 (topGen‘𝐵)(𝑋 = 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = 𝑣)))
113101, 112sylibrd 249 . 2 ((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) → (∀𝑦 ∈ 𝒫 𝐵(𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧) → (topGen‘𝐵) ∈ Comp))
11419, 113impbid 202 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 196  wa 383   = wceq 1632  wex 1853  wcel 2139  wral 3050  wrex 3051  {crab 3054  cin 3714  wss 3715  𝒫 cpw 4302   cuni 4588   ciun 4672  ran crn 5267   Fn wfn 6044  wf 6045  ontowfo 6047  cfv 6049  Fincfn 8121  topGenctg 16300  Topctop 20900  TopBasesctb 20951  Compccmp 21391
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-om 7231  df-1o 7729  df-er 7911  df-en 8122  df-dom 8123  df-fin 8125  df-topgen 16306  df-top 20901  df-bases 20952  df-cmp 21392
This theorem is referenced by: (None)
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