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Theorem tgcmp 21484
Description: A topology generated by a basis is compact iff open covers drawn from the basis have finite subcovers. (See also alexsub 22128, 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 2765 . . . . 5 (topGen‘𝐵) = (topGen‘𝐵)
21iscmp 21471 . . . 4 ((topGen‘𝐵) ∈ Comp ↔ ((topGen‘𝐵) ∈ Top ∧ ∀𝑦 ∈ 𝒫 (topGen‘𝐵)( (topGen‘𝐵) = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) (topGen‘𝐵) = 𝑧)))
32simprbi 490 . . 3 ((topGen‘𝐵) ∈ Comp → ∀𝑦 ∈ 𝒫 (topGen‘𝐵)( (topGen‘𝐵) = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) (topGen‘𝐵) = 𝑧))
4 unitg 21051 . . . . . . . 8 (𝐵 ∈ TopBases → (topGen‘𝐵) = 𝐵)
5 eqtr3 2786 . . . . . . . 8 (( (topGen‘𝐵) = 𝐵𝑋 = 𝐵) → (topGen‘𝐵) = 𝑋)
64, 5sylan 575 . . . . . . 7 ((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) → (topGen‘𝐵) = 𝑋)
76eqeq1d 2767 . . . . . 6 ((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) → ( (topGen‘𝐵) = 𝑦𝑋 = 𝑦))
86eqeq1d 2767 . . . . . . 7 ((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) → ( (topGen‘𝐵) = 𝑧𝑋 = 𝑧))
98rexbidv 3199 . . . . . 6 ((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) → (∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) (topGen‘𝐵) = 𝑧 ↔ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧))
107, 9imbi12d 335 . . . . 5 ((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) → (( (topGen‘𝐵) = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) (topGen‘𝐵) = 𝑧) ↔ (𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧)))
1110ralbidv 3133 . . . 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 21050 . . . . . . 7 (𝐵 ∈ TopBases → 𝐵 ⊆ (topGen‘𝐵))
1312adantr 472 . . . . . 6 ((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) → 𝐵 ⊆ (topGen‘𝐵))
14 sspwb 5073 . . . . . 6 (𝐵 ⊆ (topGen‘𝐵) ↔ 𝒫 𝐵 ⊆ 𝒫 (topGen‘𝐵))
1513, 14sylib 209 . . . . 5 ((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) → 𝒫 𝐵 ⊆ 𝒫 (topGen‘𝐵))
16 ssralv 3826 . . . . 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 231 . . 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 4325 . . . . 5 (𝑢 ∈ 𝒫 (topGen‘𝐵) → 𝑢 ⊆ (topGen‘𝐵))
21 simprr 789 . . . . . . . . . . 11 (((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) → 𝑋 = 𝑢)
22 simprl 787 . . . . . . . . . . . . . . . . . 18 (((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) → 𝑢 ⊆ (topGen‘𝐵))
2322sselda 3761 . . . . . . . . . . . . . . . . 17 ((((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) ∧ 𝑡𝑢) → 𝑡 ∈ (topGen‘𝐵))
2423adantrr 708 . . . . . . . . . . . . . . . 16 ((((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) ∧ (𝑡𝑢𝑦𝑡)) → 𝑡 ∈ (topGen‘𝐵))
25 simprr 789 . . . . . . . . . . . . . . . 16 ((((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) ∧ (𝑡𝑢𝑦𝑡)) → 𝑦𝑡)
26 tg2 21049 . . . . . . . . . . . . . . . 16 ((𝑡 ∈ (topGen‘𝐵) ∧ 𝑦𝑡) → ∃𝑤𝐵 (𝑦𝑤𝑤𝑡))
2724, 25, 26syl2anc 579 . . . . . . . . . . . . . . 15 ((((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) ∧ (𝑡𝑢𝑦𝑡)) → ∃𝑤𝐵 (𝑦𝑤𝑤𝑡))
2827expr 448 . . . . . . . . . . . . . 14 ((((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) ∧ 𝑡𝑢) → (𝑦𝑡 → ∃𝑤𝐵 (𝑦𝑤𝑤𝑡)))
2928reximdva 3163 . . . . . . . . . . . . 13 (((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) → (∃𝑡𝑢 𝑦𝑡 → ∃𝑡𝑢𝑤𝐵 (𝑦𝑤𝑤𝑡)))
30 eluni2 4598 . . . . . . . . . . . . 13 (𝑦 𝑢 ↔ ∃𝑡𝑢 𝑦𝑡)
31 elunirab 4606 . . . . . . . . . . . . . 14 (𝑦 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ↔ ∃𝑤𝐵 (𝑦𝑤 ∧ ∃𝑡𝑢 𝑤𝑡))
32 r19.42v 3239 . . . . . . . . . . . . . . 15 (∃𝑡𝑢 (𝑦𝑤𝑤𝑡) ↔ (𝑦𝑤 ∧ ∃𝑡𝑢 𝑤𝑡))
3332rexbii 3188 . . . . . . . . . . . . . 14 (∃𝑤𝐵𝑡𝑢 (𝑦𝑤𝑤𝑡) ↔ ∃𝑤𝐵 (𝑦𝑤 ∧ ∃𝑡𝑢 𝑤𝑡))
34 rexcom 3246 . . . . . . . . . . . . . 14 (∃𝑤𝐵𝑡𝑢 (𝑦𝑤𝑤𝑡) ↔ ∃𝑡𝑢𝑤𝐵 (𝑦𝑤𝑤𝑡))
3531, 33, 343bitr2i 290 . . . . . . . . . . . . 13 (𝑦 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ↔ ∃𝑡𝑢𝑤𝐵 (𝑦𝑤𝑤𝑡))
3629, 30, 353imtr4g 287 . . . . . . . . . . . 12 (((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) → (𝑦 𝑢𝑦 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡}))
3736ssrdv 3767 . . . . . . . . . . 11 (((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) → 𝑢 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡})
3821, 37eqsstrd 3799 . . . . . . . . . 10 (((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) → 𝑋 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡})
39 ssrab2 3847 . . . . . . . . . . . 12 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ⊆ 𝐵
4039unissi 4619 . . . . . . . . . . 11 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ⊆ 𝐵
41 simplr 785 . . . . . . . . . . 11 (((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) → 𝑋 = 𝐵)
4240, 41syl5sseqr 3814 . . . . . . . . . 10 (((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) → {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ⊆ 𝑋)
4338, 42eqssd 3778 . . . . . . . . 9 (((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) → 𝑋 = {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡})
44 elpw2g 4985 . . . . . . . . . . . 12 (𝐵 ∈ TopBases → ({𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∈ 𝒫 𝐵 ↔ {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ⊆ 𝐵))
4544ad2antrr 717 . . . . . . . . . . 11 (((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) → ({𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∈ 𝒫 𝐵 ↔ {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ⊆ 𝐵))
4639, 45mpbiri 249 . . . . . . . . . 10 (((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) → {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∈ 𝒫 𝐵)
47 unieq 4602 . . . . . . . . . . . . 13 (𝑦 = {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} → 𝑦 = {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡})
4847eqeq2d 2775 . . . . . . . . . . . 12 (𝑦 = {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} → (𝑋 = 𝑦𝑋 = {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡}))
49 pweq 4318 . . . . . . . . . . . . . 14 (𝑦 = {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} → 𝒫 𝑦 = 𝒫 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡})
5049ineq1d 3975 . . . . . . . . . . . . 13 (𝑦 = {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} → (𝒫 𝑦 ∩ Fin) = (𝒫 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∩ Fin))
5150rexeqdv 3293 . . . . . . . . . . . 12 (𝑦 = {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} → (∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧 ↔ ∃𝑧 ∈ (𝒫 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∩ Fin)𝑋 = 𝑧))
5248, 51imbi12d 335 . . . . . . . . . . 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 3457 . . . . . . . . . 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 8475 . . . . . . . . . . . . 13 (𝑧 ∈ (𝒫 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∩ Fin) ↔ (𝑧 ⊆ {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∧ 𝑧 ∈ Fin))
5756simprbi 490 . . . . . . . . . . . 12 (𝑧 ∈ (𝒫 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∩ Fin) → 𝑧 ∈ Fin)
5857ad2antrl 719 . . . . . . . . . . 11 ((((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) ∧ (𝑧 ∈ (𝒫 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∩ Fin) ∧ 𝑋 = 𝑧)) → 𝑧 ∈ Fin)
5956simplbi 491 . . . . . . . . . . . . 13 (𝑧 ∈ (𝒫 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∩ Fin) → 𝑧 ⊆ {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡})
6059ad2antrl 719 . . . . . . . . . . . 12 ((((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) ∧ (𝑧 ∈ (𝒫 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∩ Fin) ∧ 𝑋 = 𝑧)) → 𝑧 ⊆ {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡})
61 ssrab 3840 . . . . . . . . . . . . 13 (𝑧 ⊆ {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ↔ (𝑧𝐵 ∧ ∀𝑤𝑧𝑡𝑢 𝑤𝑡))
6261simprbi 490 . . . . . . . . . . . 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 3787 . . . . . . . . . . . 12 (𝑡 = (𝑓𝑤) → (𝑤𝑡𝑤 ⊆ (𝑓𝑤)))
6564ac6sfi 8411 . . . . . . . . . . 11 ((𝑧 ∈ Fin ∧ ∀𝑤𝑧𝑡𝑢 𝑤𝑡) → ∃𝑓(𝑓:𝑧𝑢 ∧ ∀𝑤𝑧 𝑤 ⊆ (𝑓𝑤)))
6658, 63, 65syl2anc 579 . . . . . . . . . 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 6229 . . . . . . . . . . . . 13 (𝑓:𝑧𝑢 → ran 𝑓𝑢)
6867ad2antrl 719 . . . . . . . . . . . 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 6223 . . . . . . . . . . . . . . 15 (𝑓:𝑧𝑢𝑓 Fn 𝑧)
70 dffn4 6304 . . . . . . . . . . . . . . 15 (𝑓 Fn 𝑧𝑓:𝑧onto→ran 𝑓)
7169, 70sylib 209 . . . . . . . . . . . . . 14 (𝑓:𝑧𝑢𝑓:𝑧onto→ran 𝑓)
7271adantr 472 . . . . . . . . . . . . 13 ((𝑓:𝑧𝑢 ∧ ∀𝑤𝑧 𝑤 ⊆ (𝑓𝑤)) → 𝑓:𝑧onto→ran 𝑓)
73 fofi 8459 . . . . . . . . . . . . 13 ((𝑧 ∈ Fin ∧ 𝑓:𝑧onto→ran 𝑓) → ran 𝑓 ∈ Fin)
7458, 72, 73syl2an 589 . . . . . . . . . . . 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 8475 . . . . . . . . . . . 12 (ran 𝑓 ∈ (𝒫 𝑢 ∩ Fin) ↔ (ran 𝑓𝑢 ∧ ran 𝑓 ∈ Fin))
7668, 74, 75sylanbrc 578 . . . . . . . . . . 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 796 . . . . . . . . . . . . 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 4729 . . . . . . . . . . . . . . . 16 𝑧 = 𝑤𝑧 𝑤
79 ss2iun 4692 . . . . . . . . . . . . . . . 16 (∀𝑤𝑧 𝑤 ⊆ (𝑓𝑤) → 𝑤𝑧 𝑤 𝑤𝑧 (𝑓𝑤))
8078, 79syl5eqss 3809 . . . . . . . . . . . . . . 15 (∀𝑤𝑧 𝑤 ⊆ (𝑓𝑤) → 𝑧 𝑤𝑧 (𝑓𝑤))
8180ad2antll 720 . . . . . . . . . . . . . 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 6697 . . . . . . . . . . . . . . . 16 (𝑓 Fn 𝑧 𝑤𝑧 (𝑓𝑤) = ran 𝑓)
8369, 82syl 17 . . . . . . . . . . . . . . 15 (𝑓:𝑧𝑢 𝑤𝑧 (𝑓𝑤) = ran 𝑓)
8483ad2antrl 719 . . . . . . . . . . . . . 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 3801 . . . . . . . . . . . . 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 3799 . . . . . . . . . . . 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 4620 . . . . . . . . . . . . 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 717 . . . . . . . . . . . . 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 3802 . . . . . . . . . . . 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 3778 . . . . . . . . . . 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 4602 . . . . . . . . . . . 12 (𝑣 = ran 𝑓 𝑣 = ran 𝑓)
9291rspceeqv 3479 . . . . . . . . . . 11 ((ran 𝑓 ∈ (𝒫 𝑢 ∩ Fin) ∧ 𝑋 = ran 𝑓) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = 𝑣)
9376, 90, 92syl2anc 579 . . . . . . . . . 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)
9466, 93exlimddv 2030 . . . . . . . . 9 ((((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) ∧ (𝑧 ∈ (𝒫 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∩ Fin) ∧ 𝑋 = 𝑧)) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = 𝑣)
9594rexlimdvaa 3179 . . . . . . . 8 (((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = 𝑢)) → (∃𝑧 ∈ (𝒫 {𝑤𝐵 ∣ ∃𝑡𝑢 𝑤𝑡} ∩ Fin)𝑋 = 𝑧 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = 𝑣))
9655, 95syld 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))
9796expr 448 . . . . . 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)))
9897com23 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)))
9920, 98sylan2 586 . . . 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)))
10099ralrimdva 3116 . . 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)))
101 tgcl 21053 . . . . . 6 (𝐵 ∈ TopBases → (topGen‘𝐵) ∈ Top)
102101adantr 472 . . . . 5 ((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) → (topGen‘𝐵) ∈ Top)
1031iscmp 21471 . . . . . 6 ((topGen‘𝐵) ∈ Comp ↔ ((topGen‘𝐵) ∈ Top ∧ ∀𝑢 ∈ 𝒫 (topGen‘𝐵)( (topGen‘𝐵) = 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin) (topGen‘𝐵) = 𝑣)))
104103baib 531 . . . . 5 ((topGen‘𝐵) ∈ Top → ((topGen‘𝐵) ∈ Comp ↔ ∀𝑢 ∈ 𝒫 (topGen‘𝐵)( (topGen‘𝐵) = 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin) (topGen‘𝐵) = 𝑣)))
105102, 104syl 17 . . . 4 ((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) → ((topGen‘𝐵) ∈ Comp ↔ ∀𝑢 ∈ 𝒫 (topGen‘𝐵)( (topGen‘𝐵) = 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin) (topGen‘𝐵) = 𝑣)))
1066eqeq1d 2767 . . . . . 6 ((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) → ( (topGen‘𝐵) = 𝑢𝑋 = 𝑢))
1076eqeq1d 2767 . . . . . . 7 ((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) → ( (topGen‘𝐵) = 𝑣𝑋 = 𝑣))
108107rexbidv 3199 . . . . . 6 ((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) → (∃𝑣 ∈ (𝒫 𝑢 ∩ Fin) (topGen‘𝐵) = 𝑣 ↔ ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = 𝑣))
109106, 108imbi12d 335 . . . . 5 ((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) → (( (topGen‘𝐵) = 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin) (topGen‘𝐵) = 𝑣) ↔ (𝑋 = 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = 𝑣)))
110109ralbidv 3133 . . . 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)))
111105, 110bitrd 270 . . 3 ((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) → ((topGen‘𝐵) ∈ Comp ↔ ∀𝑢 ∈ 𝒫 (topGen‘𝐵)(𝑋 = 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = 𝑣)))
112100, 111sylibrd 250 . 2 ((𝐵 ∈ TopBases ∧ 𝑋 = 𝐵) → (∀𝑦 ∈ 𝒫 𝐵(𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧) → (topGen‘𝐵) ∈ Comp))
11319, 112impbid 203 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 197  wa 384   = wceq 1652  wex 1874  wcel 2155  wral 3055  wrex 3056  {crab 3059  cin 3731  wss 3732  𝒫 cpw 4315   cuni 4594   ciun 4676  ran crn 5278   Fn wfn 6063  wf 6064  ontowfo 6066  cfv 6068  Fincfn 8160  topGenctg 16364  Topctop 20977  TopBasesctb 21029  Compccmp 21469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-om 7264  df-1o 7764  df-er 7947  df-en 8161  df-dom 8162  df-fin 8164  df-topgen 16370  df-top 20978  df-bases 21030  df-cmp 21470
This theorem is referenced by: (None)
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