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Theorem topjoin 32687
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 20940 . . . . . . 7 (𝑘 ∈ (TopOn‘𝑋) → 𝑘 ∈ Top)
21ad2antrl 766 . . . . . 6 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ (𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗𝑆 𝑗𝑘)) → 𝑘 ∈ Top)
3 toponmax 20952 . . . . . . . . 9 (𝑘 ∈ (TopOn‘𝑋) → 𝑋𝑘)
43ad2antrl 766 . . . . . . . 8 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ (𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗𝑆 𝑗𝑘)) → 𝑋𝑘)
54snssd 4485 . . . . . . 7 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ (𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗𝑆 𝑗𝑘)) → {𝑋} ⊆ 𝑘)
6 simprr 813 . . . . . . . 8 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ (𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗𝑆 𝑗𝑘)) → ∀𝑗𝑆 𝑗𝑘)
7 unissb 4621 . . . . . . . 8 ( 𝑆𝑘 ↔ ∀𝑗𝑆 𝑗𝑘)
86, 7sylibr 224 . . . . . . 7 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ (𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗𝑆 𝑗𝑘)) → 𝑆𝑘)
95, 8unssd 3932 . . . . . 6 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ (𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗𝑆 𝑗𝑘)) → ({𝑋} ∪ 𝑆) ⊆ 𝑘)
10 tgfiss 21017 . . . . . 6 ((𝑘 ∈ Top ∧ ({𝑋} ∪ 𝑆) ⊆ 𝑘) → (topGen‘(fi‘({𝑋} ∪ 𝑆))) ⊆ 𝑘)
112, 9, 10syl2anc 696 . . . . 5 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ (𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗𝑆 𝑗𝑘)) → (topGen‘(fi‘({𝑋} ∪ 𝑆))) ⊆ 𝑘)
1211expr 644 . . . 4 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑘 ∈ (TopOn‘𝑋)) → (∀𝑗𝑆 𝑗𝑘 → (topGen‘(fi‘({𝑋} ∪ 𝑆))) ⊆ 𝑘))
1312ralrimiva 3104 . . 3 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → ∀𝑘 ∈ (TopOn‘𝑋)(∀𝑗𝑆 𝑗𝑘 → (topGen‘(fi‘({𝑋} ∪ 𝑆))) ⊆ 𝑘))
14 ssintrab 4652 . . 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 21003 . . . . . 6 (fi‘({𝑋} ∪ 𝑆)) ∈ TopBases
17 tgtopon 20997 . . . . . 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 4608 . . . . . . . 8 ({𝑋} ∪ 𝑆) = ( {𝑋} ∪ 𝑆)
20 unisng 4604 . . . . . . . . . 10 (𝑋𝑉 {𝑋} = 𝑋)
2120adantr 472 . . . . . . . . 9 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → {𝑋} = 𝑋)
2221uneq1d 3909 . . . . . . . 8 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → ( {𝑋} ∪ 𝑆) = (𝑋 𝑆))
2319, 22syl5req 2807 . . . . . . 7 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → (𝑋 𝑆) = ({𝑋} ∪ 𝑆))
24 simpr 479 . . . . . . . . . . 11 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → 𝑆 ⊆ (TopOn‘𝑋))
25 toponuni 20941 . . . . . . . . . . . . . . 15 (𝑘 ∈ (TopOn‘𝑋) → 𝑋 = 𝑘)
26 eqimss2 3799 . . . . . . . . . . . . . . 15 (𝑋 = 𝑘 𝑘𝑋)
2725, 26syl 17 . . . . . . . . . . . . . 14 (𝑘 ∈ (TopOn‘𝑋) → 𝑘𝑋)
28 sspwuni 4763 . . . . . . . . . . . . . 14 (𝑘 ⊆ 𝒫 𝑋 𝑘𝑋)
2927, 28sylibr 224 . . . . . . . . . . . . 13 (𝑘 ∈ (TopOn‘𝑋) → 𝑘 ⊆ 𝒫 𝑋)
30 selpw 4309 . . . . . . . . . . . . 13 (𝑘 ∈ 𝒫 𝒫 𝑋𝑘 ⊆ 𝒫 𝑋)
3129, 30sylibr 224 . . . . . . . . . . . 12 (𝑘 ∈ (TopOn‘𝑋) → 𝑘 ∈ 𝒫 𝒫 𝑋)
3231ssriv 3748 . . . . . . . . . . 11 (TopOn‘𝑋) ⊆ 𝒫 𝒫 𝑋
3324, 32syl6ss 3756 . . . . . . . . . 10 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → 𝑆 ⊆ 𝒫 𝒫 𝑋)
34 sspwuni 4763 . . . . . . . . . 10 (𝑆 ⊆ 𝒫 𝒫 𝑋 𝑆 ⊆ 𝒫 𝑋)
3533, 34sylib 208 . . . . . . . . 9 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → 𝑆 ⊆ 𝒫 𝑋)
36 sspwuni 4763 . . . . . . . . 9 ( 𝑆 ⊆ 𝒫 𝑋 𝑆𝑋)
3735, 36sylib 208 . . . . . . . 8 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → 𝑆𝑋)
38 ssequn2 3929 . . . . . . . 8 ( 𝑆𝑋 ↔ (𝑋 𝑆) = 𝑋)
3937, 38sylib 208 . . . . . . 7 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → (𝑋 𝑆) = 𝑋)
40 snex 5057 . . . . . . . . 9 {𝑋} ∈ V
41 fvex 6363 . . . . . . . . . . . 12 (TopOn‘𝑋) ∈ V
4241ssex 4954 . . . . . . . . . . 11 (𝑆 ⊆ (TopOn‘𝑋) → 𝑆 ∈ V)
4342adantl 473 . . . . . . . . . 10 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → 𝑆 ∈ V)
44 uniexg 7121 . . . . . . . . . 10 (𝑆 ∈ V → 𝑆 ∈ V)
4543, 44syl 17 . . . . . . . . 9 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → 𝑆 ∈ V)
46 unexg 7125 . . . . . . . . 9 (({𝑋} ∈ V ∧ 𝑆 ∈ V) → ({𝑋} ∪ 𝑆) ∈ V)
4740, 45, 46sylancr 698 . . . . . . . 8 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → ({𝑋} ∪ 𝑆) ∈ V)
48 fiuni 8501 . . . . . . . 8 (({𝑋} ∪ 𝑆) ∈ V → ({𝑋} ∪ 𝑆) = (fi‘({𝑋} ∪ 𝑆)))
4947, 48syl 17 . . . . . . 7 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → ({𝑋} ∪ 𝑆) = (fi‘({𝑋} ∪ 𝑆)))
5023, 39, 493eqtr3d 2802 . . . . . 6 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → 𝑋 = (fi‘({𝑋} ∪ 𝑆)))
5150fveq2d 6357 . . . . 5 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → (TopOn‘𝑋) = (TopOn‘ (fi‘({𝑋} ∪ 𝑆))))
5218, 51syl5eleqr 2846 . . . 4 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → (topGen‘(fi‘({𝑋} ∪ 𝑆))) ∈ (TopOn‘𝑋))
53 elssuni 4619 . . . . . . . 8 (𝑗𝑆𝑗 𝑆)
54 ssun2 3920 . . . . . . . 8 𝑆 ⊆ ({𝑋} ∪ 𝑆)
5553, 54syl6ss 3756 . . . . . . 7 (𝑗𝑆𝑗 ⊆ ({𝑋} ∪ 𝑆))
56 ssfii 8492 . . . . . . . 8 (({𝑋} ∪ 𝑆) ∈ V → ({𝑋} ∪ 𝑆) ⊆ (fi‘({𝑋} ∪ 𝑆)))
5747, 56syl 17 . . . . . . 7 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → ({𝑋} ∪ 𝑆) ⊆ (fi‘({𝑋} ∪ 𝑆)))
5855, 57sylan9ssr 3758 . . . . . 6 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑗𝑆) → 𝑗 ⊆ (fi‘({𝑋} ∪ 𝑆)))
59 bastg 20992 . . . . . . 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 3756 . . . . 5 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑗𝑆) → 𝑗 ⊆ (topGen‘(fi‘({𝑋} ∪ 𝑆))))
6261ralrimiva 3104 . . . 4 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → ∀𝑗𝑆 𝑗 ⊆ (topGen‘(fi‘({𝑋} ∪ 𝑆))))
63 sseq2 3768 . . . . . 6 (𝑘 = (topGen‘(fi‘({𝑋} ∪ 𝑆))) → (𝑗𝑘𝑗 ⊆ (topGen‘(fi‘({𝑋} ∪ 𝑆)))))
6463ralbidv 3124 . . . . 5 (𝑘 = (topGen‘(fi‘({𝑋} ∪ 𝑆))) → (∀𝑗𝑆 𝑗𝑘 ↔ ∀𝑗𝑆 𝑗 ⊆ (topGen‘(fi‘({𝑋} ∪ 𝑆)))))
6564elrab 3504 . . . 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 701 . . 3 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → (topGen‘(fi‘({𝑋} ∪ 𝑆))) ∈ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑗𝑘})
67 intss1 4644 . . 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 3761 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 1632  wcel 2139  wral 3050  {crab 3054  Vcvv 3340  cun 3713  wss 3715  𝒫 cpw 4302  {csn 4321   cuni 4588   cint 4627  cfv 6049  ficfi 8483  topGenctg 16320  Topctop 20920  TopOnctopon 20937  TopBasesctb 20971
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 7115
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-csb 3675  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-int 4628  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-pred 5841  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-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-oadd 7734  df-er 7913  df-en 8124  df-fin 8127  df-fi 8484  df-topgen 16326  df-top 20921  df-topon 20938  df-bases 20972
This theorem is referenced by: (None)
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