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Theorem topjoin 32002
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 20641 . . . . . . 7 (𝑘 ∈ (TopOn‘𝑋) → 𝑘 ∈ Top)
21ad2antrl 763 . . . . . 6 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ (𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗𝑆 𝑗𝑘)) → 𝑘 ∈ Top)
3 toponmax 20643 . . . . . . . . 9 (𝑘 ∈ (TopOn‘𝑋) → 𝑋𝑘)
43ad2antrl 763 . . . . . . . 8 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ (𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗𝑆 𝑗𝑘)) → 𝑋𝑘)
54snssd 4309 . . . . . . 7 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ (𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗𝑆 𝑗𝑘)) → {𝑋} ⊆ 𝑘)
6 simprr 795 . . . . . . . 8 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ (𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗𝑆 𝑗𝑘)) → ∀𝑗𝑆 𝑗𝑘)
7 unissb 4435 . . . . . . . 8 ( 𝑆𝑘 ↔ ∀𝑗𝑆 𝑗𝑘)
86, 7sylibr 224 . . . . . . 7 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ (𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗𝑆 𝑗𝑘)) → 𝑆𝑘)
95, 8unssd 3767 . . . . . 6 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ (𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗𝑆 𝑗𝑘)) → ({𝑋} ∪ 𝑆) ⊆ 𝑘)
10 tgfiss 20706 . . . . . 6 ((𝑘 ∈ Top ∧ ({𝑋} ∪ 𝑆) ⊆ 𝑘) → (topGen‘(fi‘({𝑋} ∪ 𝑆))) ⊆ 𝑘)
112, 9, 10syl2anc 692 . . . . 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 2960 . . 3 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → ∀𝑘 ∈ (TopOn‘𝑋)(∀𝑗𝑆 𝑗𝑘 → (topGen‘(fi‘({𝑋} ∪ 𝑆))) ⊆ 𝑘))
14 ssintrab 4465 . . 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 20692 . . . . . 6 (fi‘({𝑋} ∪ 𝑆)) ∈ TopBases
17 tgtopon 20686 . . . . . 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 4422 . . . . . . . 8 ({𝑋} ∪ 𝑆) = ( {𝑋} ∪ 𝑆)
20 unisng 4418 . . . . . . . . . 10 (𝑋𝑉 {𝑋} = 𝑋)
2120adantr 481 . . . . . . . . 9 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → {𝑋} = 𝑋)
2221uneq1d 3744 . . . . . . . 8 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → ( {𝑋} ∪ 𝑆) = (𝑋 𝑆))
2319, 22syl5req 2668 . . . . . . 7 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → (𝑋 𝑆) = ({𝑋} ∪ 𝑆))
24 simpr 477 . . . . . . . . . . 11 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → 𝑆 ⊆ (TopOn‘𝑋))
25 toponuni 20642 . . . . . . . . . . . . . . 15 (𝑘 ∈ (TopOn‘𝑋) → 𝑋 = 𝑘)
26 eqimss2 3637 . . . . . . . . . . . . . . 15 (𝑋 = 𝑘 𝑘𝑋)
2725, 26syl 17 . . . . . . . . . . . . . 14 (𝑘 ∈ (TopOn‘𝑋) → 𝑘𝑋)
28 sspwuni 4577 . . . . . . . . . . . . . 14 (𝑘 ⊆ 𝒫 𝑋 𝑘𝑋)
2927, 28sylibr 224 . . . . . . . . . . . . 13 (𝑘 ∈ (TopOn‘𝑋) → 𝑘 ⊆ 𝒫 𝑋)
30 selpw 4137 . . . . . . . . . . . . 13 (𝑘 ∈ 𝒫 𝒫 𝑋𝑘 ⊆ 𝒫 𝑋)
3129, 30sylibr 224 . . . . . . . . . . . 12 (𝑘 ∈ (TopOn‘𝑋) → 𝑘 ∈ 𝒫 𝒫 𝑋)
3231ssriv 3587 . . . . . . . . . . 11 (TopOn‘𝑋) ⊆ 𝒫 𝒫 𝑋
3324, 32syl6ss 3595 . . . . . . . . . 10 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → 𝑆 ⊆ 𝒫 𝒫 𝑋)
34 sspwuni 4577 . . . . . . . . . 10 (𝑆 ⊆ 𝒫 𝒫 𝑋 𝑆 ⊆ 𝒫 𝑋)
3533, 34sylib 208 . . . . . . . . 9 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → 𝑆 ⊆ 𝒫 𝑋)
36 sspwuni 4577 . . . . . . . . 9 ( 𝑆 ⊆ 𝒫 𝑋 𝑆𝑋)
3735, 36sylib 208 . . . . . . . 8 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → 𝑆𝑋)
38 ssequn2 3764 . . . . . . . 8 ( 𝑆𝑋 ↔ (𝑋 𝑆) = 𝑋)
3937, 38sylib 208 . . . . . . 7 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → (𝑋 𝑆) = 𝑋)
40 snex 4869 . . . . . . . . 9 {𝑋} ∈ V
41 fvex 6158 . . . . . . . . . . . 12 (TopOn‘𝑋) ∈ V
4241ssex 4762 . . . . . . . . . . 11 (𝑆 ⊆ (TopOn‘𝑋) → 𝑆 ∈ V)
4342adantl 482 . . . . . . . . . 10 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → 𝑆 ∈ V)
44 uniexg 6908 . . . . . . . . . 10 (𝑆 ∈ V → 𝑆 ∈ V)
4543, 44syl 17 . . . . . . . . 9 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → 𝑆 ∈ V)
46 unexg 6912 . . . . . . . . 9 (({𝑋} ∈ V ∧ 𝑆 ∈ V) → ({𝑋} ∪ 𝑆) ∈ V)
4740, 45, 46sylancr 694 . . . . . . . 8 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → ({𝑋} ∪ 𝑆) ∈ V)
48 fiuni 8278 . . . . . . . 8 (({𝑋} ∪ 𝑆) ∈ V → ({𝑋} ∪ 𝑆) = (fi‘({𝑋} ∪ 𝑆)))
4947, 48syl 17 . . . . . . 7 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → ({𝑋} ∪ 𝑆) = (fi‘({𝑋} ∪ 𝑆)))
5023, 39, 493eqtr3d 2663 . . . . . 6 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → 𝑋 = (fi‘({𝑋} ∪ 𝑆)))
5150fveq2d 6152 . . . . 5 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → (TopOn‘𝑋) = (TopOn‘ (fi‘({𝑋} ∪ 𝑆))))
5218, 51syl5eleqr 2705 . . . 4 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → (topGen‘(fi‘({𝑋} ∪ 𝑆))) ∈ (TopOn‘𝑋))
53 elssuni 4433 . . . . . . . 8 (𝑗𝑆𝑗 𝑆)
54 ssun2 3755 . . . . . . . 8 𝑆 ⊆ ({𝑋} ∪ 𝑆)
5553, 54syl6ss 3595 . . . . . . 7 (𝑗𝑆𝑗 ⊆ ({𝑋} ∪ 𝑆))
56 ssfii 8269 . . . . . . . 8 (({𝑋} ∪ 𝑆) ∈ V → ({𝑋} ∪ 𝑆) ⊆ (fi‘({𝑋} ∪ 𝑆)))
5747, 56syl 17 . . . . . . 7 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → ({𝑋} ∪ 𝑆) ⊆ (fi‘({𝑋} ∪ 𝑆)))
5855, 57sylan9ssr 3597 . . . . . 6 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑗𝑆) → 𝑗 ⊆ (fi‘({𝑋} ∪ 𝑆)))
59 bastg 20681 . . . . . . 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 3595 . . . . 5 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑗𝑆) → 𝑗 ⊆ (topGen‘(fi‘({𝑋} ∪ 𝑆))))
6261ralrimiva 2960 . . . 4 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → ∀𝑗𝑆 𝑗 ⊆ (topGen‘(fi‘({𝑋} ∪ 𝑆))))
63 sseq2 3606 . . . . . 6 (𝑘 = (topGen‘(fi‘({𝑋} ∪ 𝑆))) → (𝑗𝑘𝑗 ⊆ (topGen‘(fi‘({𝑋} ∪ 𝑆)))))
6463ralbidv 2980 . . . . 5 (𝑘 = (topGen‘(fi‘({𝑋} ∪ 𝑆))) → (∀𝑗𝑆 𝑗𝑘 ↔ ∀𝑗𝑆 𝑗 ⊆ (topGen‘(fi‘({𝑋} ∪ 𝑆)))))
6564elrab 3346 . . . 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 697 . . 3 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → (topGen‘(fi‘({𝑋} ∪ 𝑆))) ∈ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑗𝑘})
67 intss1 4457 . . 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 3600 1 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → (topGen‘(fi‘({𝑋} ∪ 𝑆))) = {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑗𝑘})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wral 2907  {crab 2911  Vcvv 3186  cun 3553  wss 3555  𝒫 cpw 4130  {csn 4148   cuni 4402   cint 4440  cfv 5847  ficfi 8260  topGenctg 16019  Topctop 20617  TopOnctopon 20618  TopBasesctb 20620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-en 7900  df-fin 7903  df-fi 8261  df-topgen 16025  df-top 20621  df-bases 20622  df-topon 20623
This theorem is referenced by: (None)
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