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Theorem topjoin 32485
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 20766 . . . . . . 7 (𝑘 ∈ (TopOn‘𝑋) → 𝑘 ∈ Top)
21ad2antrl 764 . . . . . 6 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ (𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗𝑆 𝑗𝑘)) → 𝑘 ∈ Top)
3 toponmax 20778 . . . . . . . . 9 (𝑘 ∈ (TopOn‘𝑋) → 𝑋𝑘)
43ad2antrl 764 . . . . . . . 8 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ (𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗𝑆 𝑗𝑘)) → 𝑋𝑘)
54snssd 4372 . . . . . . 7 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ (𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗𝑆 𝑗𝑘)) → {𝑋} ⊆ 𝑘)
6 simprr 811 . . . . . . . 8 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ (𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗𝑆 𝑗𝑘)) → ∀𝑗𝑆 𝑗𝑘)
7 unissb 4501 . . . . . . . 8 ( 𝑆𝑘 ↔ ∀𝑗𝑆 𝑗𝑘)
86, 7sylibr 224 . . . . . . 7 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ (𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗𝑆 𝑗𝑘)) → 𝑆𝑘)
95, 8unssd 3822 . . . . . 6 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ (𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗𝑆 𝑗𝑘)) → ({𝑋} ∪ 𝑆) ⊆ 𝑘)
10 tgfiss 20843 . . . . . 6 ((𝑘 ∈ Top ∧ ({𝑋} ∪ 𝑆) ⊆ 𝑘) → (topGen‘(fi‘({𝑋} ∪ 𝑆))) ⊆ 𝑘)
112, 9, 10syl2anc 694 . . . . 5 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ (𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗𝑆 𝑗𝑘)) → (topGen‘(fi‘({𝑋} ∪ 𝑆))) ⊆ 𝑘)
1211expr 642 . . . 4 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑘 ∈ (TopOn‘𝑋)) → (∀𝑗𝑆 𝑗𝑘 → (topGen‘(fi‘({𝑋} ∪ 𝑆))) ⊆ 𝑘))
1312ralrimiva 2995 . . 3 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → ∀𝑘 ∈ (TopOn‘𝑋)(∀𝑗𝑆 𝑗𝑘 → (topGen‘(fi‘({𝑋} ∪ 𝑆))) ⊆ 𝑘))
14 ssintrab 4532 . . 3 ((topGen‘(fi‘({𝑋} ∪ 𝑆))) ⊆ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑗𝑘} ↔ ∀𝑘 ∈ (TopOn‘𝑋)(∀𝑗𝑆 𝑗𝑘 → (topGen‘(fi‘({𝑋} ∪ 𝑆))) ⊆ 𝑘))
1513, 14sylibr 224 . 2 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → (topGen‘(fi‘({𝑋} ∪ 𝑆))) ⊆ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑗𝑘})
16 fibas 20829 . . . . . 6 (fi‘({𝑋} ∪ 𝑆)) ∈ TopBases
17 tgtopon 20823 . . . . . 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 4488 . . . . . . . 8 ({𝑋} ∪ 𝑆) = ( {𝑋} ∪ 𝑆)
20 unisng 4484 . . . . . . . . . 10 (𝑋𝑉 {𝑋} = 𝑋)
2120adantr 480 . . . . . . . . 9 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → {𝑋} = 𝑋)
2221uneq1d 3799 . . . . . . . 8 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → ( {𝑋} ∪ 𝑆) = (𝑋 𝑆))
2319, 22syl5req 2698 . . . . . . 7 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → (𝑋 𝑆) = ({𝑋} ∪ 𝑆))
24 simpr 476 . . . . . . . . . . 11 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → 𝑆 ⊆ (TopOn‘𝑋))
25 toponuni 20767 . . . . . . . . . . . . . . 15 (𝑘 ∈ (TopOn‘𝑋) → 𝑋 = 𝑘)
26 eqimss2 3691 . . . . . . . . . . . . . . 15 (𝑋 = 𝑘 𝑘𝑋)
2725, 26syl 17 . . . . . . . . . . . . . 14 (𝑘 ∈ (TopOn‘𝑋) → 𝑘𝑋)
28 sspwuni 4643 . . . . . . . . . . . . . 14 (𝑘 ⊆ 𝒫 𝑋 𝑘𝑋)
2927, 28sylibr 224 . . . . . . . . . . . . 13 (𝑘 ∈ (TopOn‘𝑋) → 𝑘 ⊆ 𝒫 𝑋)
30 selpw 4198 . . . . . . . . . . . . 13 (𝑘 ∈ 𝒫 𝒫 𝑋𝑘 ⊆ 𝒫 𝑋)
3129, 30sylibr 224 . . . . . . . . . . . 12 (𝑘 ∈ (TopOn‘𝑋) → 𝑘 ∈ 𝒫 𝒫 𝑋)
3231ssriv 3640 . . . . . . . . . . 11 (TopOn‘𝑋) ⊆ 𝒫 𝒫 𝑋
3324, 32syl6ss 3648 . . . . . . . . . 10 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → 𝑆 ⊆ 𝒫 𝒫 𝑋)
34 sspwuni 4643 . . . . . . . . . 10 (𝑆 ⊆ 𝒫 𝒫 𝑋 𝑆 ⊆ 𝒫 𝑋)
3533, 34sylib 208 . . . . . . . . 9 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → 𝑆 ⊆ 𝒫 𝑋)
36 sspwuni 4643 . . . . . . . . 9 ( 𝑆 ⊆ 𝒫 𝑋 𝑆𝑋)
3735, 36sylib 208 . . . . . . . 8 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → 𝑆𝑋)
38 ssequn2 3819 . . . . . . . 8 ( 𝑆𝑋 ↔ (𝑋 𝑆) = 𝑋)
3937, 38sylib 208 . . . . . . 7 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → (𝑋 𝑆) = 𝑋)
40 snex 4938 . . . . . . . . 9 {𝑋} ∈ V
41 fvex 6239 . . . . . . . . . . . 12 (TopOn‘𝑋) ∈ V
4241ssex 4835 . . . . . . . . . . 11 (𝑆 ⊆ (TopOn‘𝑋) → 𝑆 ∈ V)
4342adantl 481 . . . . . . . . . 10 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → 𝑆 ∈ V)
44 uniexg 6997 . . . . . . . . . 10 (𝑆 ∈ V → 𝑆 ∈ V)
4543, 44syl 17 . . . . . . . . 9 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → 𝑆 ∈ V)
46 unexg 7001 . . . . . . . . 9 (({𝑋} ∈ V ∧ 𝑆 ∈ V) → ({𝑋} ∪ 𝑆) ∈ V)
4740, 45, 46sylancr 696 . . . . . . . 8 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → ({𝑋} ∪ 𝑆) ∈ V)
48 fiuni 8375 . . . . . . . 8 (({𝑋} ∪ 𝑆) ∈ V → ({𝑋} ∪ 𝑆) = (fi‘({𝑋} ∪ 𝑆)))
4947, 48syl 17 . . . . . . 7 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → ({𝑋} ∪ 𝑆) = (fi‘({𝑋} ∪ 𝑆)))
5023, 39, 493eqtr3d 2693 . . . . . 6 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → 𝑋 = (fi‘({𝑋} ∪ 𝑆)))
5150fveq2d 6233 . . . . 5 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → (TopOn‘𝑋) = (TopOn‘ (fi‘({𝑋} ∪ 𝑆))))
5218, 51syl5eleqr 2737 . . . 4 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → (topGen‘(fi‘({𝑋} ∪ 𝑆))) ∈ (TopOn‘𝑋))
53 elssuni 4499 . . . . . . . 8 (𝑗𝑆𝑗 𝑆)
54 ssun2 3810 . . . . . . . 8 𝑆 ⊆ ({𝑋} ∪ 𝑆)
5553, 54syl6ss 3648 . . . . . . 7 (𝑗𝑆𝑗 ⊆ ({𝑋} ∪ 𝑆))
56 ssfii 8366 . . . . . . . 8 (({𝑋} ∪ 𝑆) ∈ V → ({𝑋} ∪ 𝑆) ⊆ (fi‘({𝑋} ∪ 𝑆)))
5747, 56syl 17 . . . . . . 7 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → ({𝑋} ∪ 𝑆) ⊆ (fi‘({𝑋} ∪ 𝑆)))
5855, 57sylan9ssr 3650 . . . . . 6 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑗𝑆) → 𝑗 ⊆ (fi‘({𝑋} ∪ 𝑆)))
59 bastg 20818 . . . . . . 7 ((fi‘({𝑋} ∪ 𝑆)) ∈ TopBases → (fi‘({𝑋} ∪ 𝑆)) ⊆ (topGen‘(fi‘({𝑋} ∪ 𝑆))))
6016, 59ax-mp 5 . . . . . 6 (fi‘({𝑋} ∪ 𝑆)) ⊆ (topGen‘(fi‘({𝑋} ∪ 𝑆)))
6158, 60syl6ss 3648 . . . . 5 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑗𝑆) → 𝑗 ⊆ (topGen‘(fi‘({𝑋} ∪ 𝑆))))
6261ralrimiva 2995 . . . 4 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → ∀𝑗𝑆 𝑗 ⊆ (topGen‘(fi‘({𝑋} ∪ 𝑆))))
63 sseq2 3660 . . . . . 6 (𝑘 = (topGen‘(fi‘({𝑋} ∪ 𝑆))) → (𝑗𝑘𝑗 ⊆ (topGen‘(fi‘({𝑋} ∪ 𝑆)))))
6463ralbidv 3015 . . . . 5 (𝑘 = (topGen‘(fi‘({𝑋} ∪ 𝑆))) → (∀𝑗𝑆 𝑗𝑘 ↔ ∀𝑗𝑆 𝑗 ⊆ (topGen‘(fi‘({𝑋} ∪ 𝑆)))))
6564elrab 3396 . . . 4 ((topGen‘(fi‘({𝑋} ∪ 𝑆))) ∈ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑗𝑘} ↔ ((topGen‘(fi‘({𝑋} ∪ 𝑆))) ∈ (TopOn‘𝑋) ∧ ∀𝑗𝑆 𝑗 ⊆ (topGen‘(fi‘({𝑋} ∪ 𝑆)))))
6652, 62, 65sylanbrc 699 . . 3 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → (topGen‘(fi‘({𝑋} ∪ 𝑆))) ∈ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑗𝑘})
67 intss1 4524 . . 3 ((topGen‘(fi‘({𝑋} ∪ 𝑆))) ∈ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑗𝑘} → {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑗𝑘} ⊆ (topGen‘(fi‘({𝑋} ∪ 𝑆))))
6866, 67syl 17 . 2 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑗𝑘} ⊆ (topGen‘(fi‘({𝑋} ∪ 𝑆))))
6915, 68eqssd 3653 1 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → (topGen‘(fi‘({𝑋} ∪ 𝑆))) = {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑗𝑘})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wral 2941  {crab 2945  Vcvv 3231  cun 3605  wss 3607  𝒫 cpw 4191  {csn 4210   cuni 4468   cint 4507  cfv 5926  ficfi 8357  topGenctg 16145  Topctop 20746  TopOnctopon 20763  TopBasesctb 20797
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-fin 8001  df-fi 8358  df-topgen 16151  df-top 20747  df-topon 20764  df-bases 20798
This theorem is referenced by: (None)
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