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Theorem topjoin 36366
Description: Two equivalent formulations of the join of a collection of topologies. (Contributed by Jeff Hankins, 6-Oct-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
Assertion
Ref Expression
topjoin ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → (topGen‘(fi‘({𝑋} ∪ 𝑆))) = {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑗𝑘})
Distinct variable groups:   𝑗,𝑘,𝑆   𝑗,𝑉,𝑘   𝑗,𝑋,𝑘

Proof of Theorem topjoin
StepHypRef Expression
1 topontop 22919 . . . . . . 7 (𝑘 ∈ (TopOn‘𝑋) → 𝑘 ∈ Top)
21ad2antrl 728 . . . . . 6 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ (𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗𝑆 𝑗𝑘)) → 𝑘 ∈ Top)
3 toponmax 22932 . . . . . . . . 9 (𝑘 ∈ (TopOn‘𝑋) → 𝑋𝑘)
43ad2antrl 728 . . . . . . . 8 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ (𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗𝑆 𝑗𝑘)) → 𝑋𝑘)
54snssd 4809 . . . . . . 7 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ (𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗𝑆 𝑗𝑘)) → {𝑋} ⊆ 𝑘)
6 simprr 773 . . . . . . . 8 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ (𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗𝑆 𝑗𝑘)) → ∀𝑗𝑆 𝑗𝑘)
7 unissb 4939 . . . . . . . 8 ( 𝑆𝑘 ↔ ∀𝑗𝑆 𝑗𝑘)
86, 7sylibr 234 . . . . . . 7 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ (𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗𝑆 𝑗𝑘)) → 𝑆𝑘)
95, 8unssd 4192 . . . . . 6 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ (𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗𝑆 𝑗𝑘)) → ({𝑋} ∪ 𝑆) ⊆ 𝑘)
10 tgfiss 22998 . . . . . 6 ((𝑘 ∈ Top ∧ ({𝑋} ∪ 𝑆) ⊆ 𝑘) → (topGen‘(fi‘({𝑋} ∪ 𝑆))) ⊆ 𝑘)
112, 9, 10syl2anc 584 . . . . 5 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ (𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗𝑆 𝑗𝑘)) → (topGen‘(fi‘({𝑋} ∪ 𝑆))) ⊆ 𝑘)
1211expr 456 . . . 4 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑘 ∈ (TopOn‘𝑋)) → (∀𝑗𝑆 𝑗𝑘 → (topGen‘(fi‘({𝑋} ∪ 𝑆))) ⊆ 𝑘))
1312ralrimiva 3146 . . 3 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → ∀𝑘 ∈ (TopOn‘𝑋)(∀𝑗𝑆 𝑗𝑘 → (topGen‘(fi‘({𝑋} ∪ 𝑆))) ⊆ 𝑘))
14 ssintrab 4971 . . 3 ((topGen‘(fi‘({𝑋} ∪ 𝑆))) ⊆ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑗𝑘} ↔ ∀𝑘 ∈ (TopOn‘𝑋)(∀𝑗𝑆 𝑗𝑘 → (topGen‘(fi‘({𝑋} ∪ 𝑆))) ⊆ 𝑘))
1513, 14sylibr 234 . 2 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → (topGen‘(fi‘({𝑋} ∪ 𝑆))) ⊆ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑗𝑘})
16 fibas 22984 . . . . . 6 (fi‘({𝑋} ∪ 𝑆)) ∈ TopBases
17 tgtopon 22978 . . . . . 6 ((fi‘({𝑋} ∪ 𝑆)) ∈ TopBases → (topGen‘(fi‘({𝑋} ∪ 𝑆))) ∈ (TopOn‘ (fi‘({𝑋} ∪ 𝑆))))
1816, 17ax-mp 5 . . . . 5 (topGen‘(fi‘({𝑋} ∪ 𝑆))) ∈ (TopOn‘ (fi‘({𝑋} ∪ 𝑆)))
19 uniun 4930 . . . . . . . 8 ({𝑋} ∪ 𝑆) = ( {𝑋} ∪ 𝑆)
20 unisng 4925 . . . . . . . . . 10 (𝑋𝑉 {𝑋} = 𝑋)
2120adantr 480 . . . . . . . . 9 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → {𝑋} = 𝑋)
2221uneq1d 4167 . . . . . . . 8 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → ( {𝑋} ∪ 𝑆) = (𝑋 𝑆))
2319, 22eqtr2id 2790 . . . . . . 7 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → (𝑋 𝑆) = ({𝑋} ∪ 𝑆))
24 simpr 484 . . . . . . . . . . 11 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → 𝑆 ⊆ (TopOn‘𝑋))
25 toponuni 22920 . . . . . . . . . . . . . . 15 (𝑘 ∈ (TopOn‘𝑋) → 𝑋 = 𝑘)
26 eqimss2 4043 . . . . . . . . . . . . . . 15 (𝑋 = 𝑘 𝑘𝑋)
2725, 26syl 17 . . . . . . . . . . . . . 14 (𝑘 ∈ (TopOn‘𝑋) → 𝑘𝑋)
28 sspwuni 5100 . . . . . . . . . . . . . 14 (𝑘 ⊆ 𝒫 𝑋 𝑘𝑋)
2927, 28sylibr 234 . . . . . . . . . . . . 13 (𝑘 ∈ (TopOn‘𝑋) → 𝑘 ⊆ 𝒫 𝑋)
30 velpw 4605 . . . . . . . . . . . . 13 (𝑘 ∈ 𝒫 𝒫 𝑋𝑘 ⊆ 𝒫 𝑋)
3129, 30sylibr 234 . . . . . . . . . . . 12 (𝑘 ∈ (TopOn‘𝑋) → 𝑘 ∈ 𝒫 𝒫 𝑋)
3231ssriv 3987 . . . . . . . . . . 11 (TopOn‘𝑋) ⊆ 𝒫 𝒫 𝑋
3324, 32sstrdi 3996 . . . . . . . . . 10 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → 𝑆 ⊆ 𝒫 𝒫 𝑋)
34 sspwuni 5100 . . . . . . . . . 10 (𝑆 ⊆ 𝒫 𝒫 𝑋 𝑆 ⊆ 𝒫 𝑋)
3533, 34sylib 218 . . . . . . . . 9 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → 𝑆 ⊆ 𝒫 𝑋)
36 sspwuni 5100 . . . . . . . . 9 ( 𝑆 ⊆ 𝒫 𝑋 𝑆𝑋)
3735, 36sylib 218 . . . . . . . 8 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → 𝑆𝑋)
38 ssequn2 4189 . . . . . . . 8 ( 𝑆𝑋 ↔ (𝑋 𝑆) = 𝑋)
3937, 38sylib 218 . . . . . . 7 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → (𝑋 𝑆) = 𝑋)
40 snex 5436 . . . . . . . . 9 {𝑋} ∈ V
41 fvex 6919 . . . . . . . . . . . 12 (TopOn‘𝑋) ∈ V
4241ssex 5321 . . . . . . . . . . 11 (𝑆 ⊆ (TopOn‘𝑋) → 𝑆 ∈ V)
4342adantl 481 . . . . . . . . . 10 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → 𝑆 ∈ V)
4443uniexd 7762 . . . . . . . . 9 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → 𝑆 ∈ V)
45 unexg 7763 . . . . . . . . 9 (({𝑋} ∈ V ∧ 𝑆 ∈ V) → ({𝑋} ∪ 𝑆) ∈ V)
4640, 44, 45sylancr 587 . . . . . . . 8 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → ({𝑋} ∪ 𝑆) ∈ V)
47 fiuni 9468 . . . . . . . 8 (({𝑋} ∪ 𝑆) ∈ V → ({𝑋} ∪ 𝑆) = (fi‘({𝑋} ∪ 𝑆)))
4846, 47syl 17 . . . . . . 7 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → ({𝑋} ∪ 𝑆) = (fi‘({𝑋} ∪ 𝑆)))
4923, 39, 483eqtr3d 2785 . . . . . 6 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → 𝑋 = (fi‘({𝑋} ∪ 𝑆)))
5049fveq2d 6910 . . . . 5 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → (TopOn‘𝑋) = (TopOn‘ (fi‘({𝑋} ∪ 𝑆))))
5118, 50eleqtrrid 2848 . . . 4 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → (topGen‘(fi‘({𝑋} ∪ 𝑆))) ∈ (TopOn‘𝑋))
52 elssuni 4937 . . . . . . . 8 (𝑗𝑆𝑗 𝑆)
53 ssun2 4179 . . . . . . . 8 𝑆 ⊆ ({𝑋} ∪ 𝑆)
5452, 53sstrdi 3996 . . . . . . 7 (𝑗𝑆𝑗 ⊆ ({𝑋} ∪ 𝑆))
55 ssfii 9459 . . . . . . . 8 (({𝑋} ∪ 𝑆) ∈ V → ({𝑋} ∪ 𝑆) ⊆ (fi‘({𝑋} ∪ 𝑆)))
5646, 55syl 17 . . . . . . 7 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → ({𝑋} ∪ 𝑆) ⊆ (fi‘({𝑋} ∪ 𝑆)))
5754, 56sylan9ssr 3998 . . . . . 6 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑗𝑆) → 𝑗 ⊆ (fi‘({𝑋} ∪ 𝑆)))
58 bastg 22973 . . . . . . 7 ((fi‘({𝑋} ∪ 𝑆)) ∈ TopBases → (fi‘({𝑋} ∪ 𝑆)) ⊆ (topGen‘(fi‘({𝑋} ∪ 𝑆))))
5916, 58ax-mp 5 . . . . . 6 (fi‘({𝑋} ∪ 𝑆)) ⊆ (topGen‘(fi‘({𝑋} ∪ 𝑆)))
6057, 59sstrdi 3996 . . . . 5 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑗𝑆) → 𝑗 ⊆ (topGen‘(fi‘({𝑋} ∪ 𝑆))))
6160ralrimiva 3146 . . . 4 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → ∀𝑗𝑆 𝑗 ⊆ (topGen‘(fi‘({𝑋} ∪ 𝑆))))
62 sseq2 4010 . . . . . 6 (𝑘 = (topGen‘(fi‘({𝑋} ∪ 𝑆))) → (𝑗𝑘𝑗 ⊆ (topGen‘(fi‘({𝑋} ∪ 𝑆)))))
6362ralbidv 3178 . . . . 5 (𝑘 = (topGen‘(fi‘({𝑋} ∪ 𝑆))) → (∀𝑗𝑆 𝑗𝑘 ↔ ∀𝑗𝑆 𝑗 ⊆ (topGen‘(fi‘({𝑋} ∪ 𝑆)))))
6463elrab 3692 . . . 4 ((topGen‘(fi‘({𝑋} ∪ 𝑆))) ∈ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑗𝑘} ↔ ((topGen‘(fi‘({𝑋} ∪ 𝑆))) ∈ (TopOn‘𝑋) ∧ ∀𝑗𝑆 𝑗 ⊆ (topGen‘(fi‘({𝑋} ∪ 𝑆)))))
6551, 61, 64sylanbrc 583 . . 3 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → (topGen‘(fi‘({𝑋} ∪ 𝑆))) ∈ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑗𝑘})
66 intss1 4963 . . 3 ((topGen‘(fi‘({𝑋} ∪ 𝑆))) ∈ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑗𝑘} → {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑗𝑘} ⊆ (topGen‘(fi‘({𝑋} ∪ 𝑆))))
6765, 66syl 17 . 2 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑗𝑘} ⊆ (topGen‘(fi‘({𝑋} ∪ 𝑆))))
6815, 67eqssd 4001 1 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → (topGen‘(fi‘({𝑋} ∪ 𝑆))) = {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑗𝑘})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wral 3061  {crab 3436  Vcvv 3480  cun 3949  wss 3951  𝒫 cpw 4600  {csn 4626   cuni 4907   cint 4946  cfv 6561  ficfi 9450  topGenctg 17482  Topctop 22899  TopOnctopon 22916  TopBasesctb 22952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-om 7888  df-1o 8506  df-2o 8507  df-en 8986  df-fin 8989  df-fi 9451  df-topgen 17488  df-top 22900  df-topon 22917  df-bases 22953
This theorem is referenced by: (None)
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