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Theorem topjoin 33597
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 21437 . . . . . . 7 (𝑘 ∈ (TopOn‘𝑋) → 𝑘 ∈ Top)
21ad2antrl 724 . . . . . 6 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ (𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗𝑆 𝑗𝑘)) → 𝑘 ∈ Top)
3 toponmax 21450 . . . . . . . . 9 (𝑘 ∈ (TopOn‘𝑋) → 𝑋𝑘)
43ad2antrl 724 . . . . . . . 8 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ (𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗𝑆 𝑗𝑘)) → 𝑋𝑘)
54snssd 4741 . . . . . . 7 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ (𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗𝑆 𝑗𝑘)) → {𝑋} ⊆ 𝑘)
6 simprr 769 . . . . . . . 8 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ (𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗𝑆 𝑗𝑘)) → ∀𝑗𝑆 𝑗𝑘)
7 unissb 4868 . . . . . . . 8 ( 𝑆𝑘 ↔ ∀𝑗𝑆 𝑗𝑘)
86, 7sylibr 235 . . . . . . 7 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ (𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗𝑆 𝑗𝑘)) → 𝑆𝑘)
95, 8unssd 4166 . . . . . 6 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ (𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗𝑆 𝑗𝑘)) → ({𝑋} ∪ 𝑆) ⊆ 𝑘)
10 tgfiss 21515 . . . . . 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 457 . . . 4 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑘 ∈ (TopOn‘𝑋)) → (∀𝑗𝑆 𝑗𝑘 → (topGen‘(fi‘({𝑋} ∪ 𝑆))) ⊆ 𝑘))
1312ralrimiva 3187 . . 3 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → ∀𝑘 ∈ (TopOn‘𝑋)(∀𝑗𝑆 𝑗𝑘 → (topGen‘(fi‘({𝑋} ∪ 𝑆))) ⊆ 𝑘))
14 ssintrab 4897 . . 3 ((topGen‘(fi‘({𝑋} ∪ 𝑆))) ⊆ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑗𝑘} ↔ ∀𝑘 ∈ (TopOn‘𝑋)(∀𝑗𝑆 𝑗𝑘 → (topGen‘(fi‘({𝑋} ∪ 𝑆))) ⊆ 𝑘))
1513, 14sylibr 235 . 2 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → (topGen‘(fi‘({𝑋} ∪ 𝑆))) ⊆ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑗𝑘})
16 fibas 21501 . . . . . 6 (fi‘({𝑋} ∪ 𝑆)) ∈ TopBases
17 tgtopon 21495 . . . . . 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 4856 . . . . . . . 8 ({𝑋} ∪ 𝑆) = ( {𝑋} ∪ 𝑆)
20 unisng 4852 . . . . . . . . . 10 (𝑋𝑉 {𝑋} = 𝑋)
2120adantr 481 . . . . . . . . 9 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → {𝑋} = 𝑋)
2221uneq1d 4142 . . . . . . . 8 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → ( {𝑋} ∪ 𝑆) = (𝑋 𝑆))
2319, 22syl5req 2874 . . . . . . 7 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → (𝑋 𝑆) = ({𝑋} ∪ 𝑆))
24 simpr 485 . . . . . . . . . . 11 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → 𝑆 ⊆ (TopOn‘𝑋))
25 toponuni 21438 . . . . . . . . . . . . . . 15 (𝑘 ∈ (TopOn‘𝑋) → 𝑋 = 𝑘)
26 eqimss2 4028 . . . . . . . . . . . . . . 15 (𝑋 = 𝑘 𝑘𝑋)
2725, 26syl 17 . . . . . . . . . . . . . 14 (𝑘 ∈ (TopOn‘𝑋) → 𝑘𝑋)
28 sspwuni 5019 . . . . . . . . . . . . . 14 (𝑘 ⊆ 𝒫 𝑋 𝑘𝑋)
2927, 28sylibr 235 . . . . . . . . . . . . 13 (𝑘 ∈ (TopOn‘𝑋) → 𝑘 ⊆ 𝒫 𝑋)
30 velpw 4550 . . . . . . . . . . . . 13 (𝑘 ∈ 𝒫 𝒫 𝑋𝑘 ⊆ 𝒫 𝑋)
3129, 30sylibr 235 . . . . . . . . . . . 12 (𝑘 ∈ (TopOn‘𝑋) → 𝑘 ∈ 𝒫 𝒫 𝑋)
3231ssriv 3975 . . . . . . . . . . 11 (TopOn‘𝑋) ⊆ 𝒫 𝒫 𝑋
3324, 32syl6ss 3983 . . . . . . . . . 10 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → 𝑆 ⊆ 𝒫 𝒫 𝑋)
34 sspwuni 5019 . . . . . . . . . 10 (𝑆 ⊆ 𝒫 𝒫 𝑋 𝑆 ⊆ 𝒫 𝑋)
3533, 34sylib 219 . . . . . . . . 9 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → 𝑆 ⊆ 𝒫 𝑋)
36 sspwuni 5019 . . . . . . . . 9 ( 𝑆 ⊆ 𝒫 𝑋 𝑆𝑋)
3735, 36sylib 219 . . . . . . . 8 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → 𝑆𝑋)
38 ssequn2 4163 . . . . . . . 8 ( 𝑆𝑋 ↔ (𝑋 𝑆) = 𝑋)
3937, 38sylib 219 . . . . . . 7 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → (𝑋 𝑆) = 𝑋)
40 snex 5328 . . . . . . . . 9 {𝑋} ∈ V
41 fvex 6680 . . . . . . . . . . . 12 (TopOn‘𝑋) ∈ V
4241ssex 5222 . . . . . . . . . . 11 (𝑆 ⊆ (TopOn‘𝑋) → 𝑆 ∈ V)
4342adantl 482 . . . . . . . . . 10 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → 𝑆 ∈ V)
44 uniexg 7457 . . . . . . . . . 10 (𝑆 ∈ V → 𝑆 ∈ V)
4543, 44syl 17 . . . . . . . . 9 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → 𝑆 ∈ V)
46 unexg 7461 . . . . . . . . 9 (({𝑋} ∈ V ∧ 𝑆 ∈ V) → ({𝑋} ∪ 𝑆) ∈ V)
4740, 45, 46sylancr 587 . . . . . . . 8 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → ({𝑋} ∪ 𝑆) ∈ V)
48 fiuni 8881 . . . . . . . 8 (({𝑋} ∪ 𝑆) ∈ V → ({𝑋} ∪ 𝑆) = (fi‘({𝑋} ∪ 𝑆)))
4947, 48syl 17 . . . . . . 7 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → ({𝑋} ∪ 𝑆) = (fi‘({𝑋} ∪ 𝑆)))
5023, 39, 493eqtr3d 2869 . . . . . 6 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → 𝑋 = (fi‘({𝑋} ∪ 𝑆)))
5150fveq2d 6671 . . . . 5 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → (TopOn‘𝑋) = (TopOn‘ (fi‘({𝑋} ∪ 𝑆))))
5218, 51eleqtrrid 2925 . . . 4 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → (topGen‘(fi‘({𝑋} ∪ 𝑆))) ∈ (TopOn‘𝑋))
53 elssuni 4866 . . . . . . . 8 (𝑗𝑆𝑗 𝑆)
54 ssun2 4153 . . . . . . . 8 𝑆 ⊆ ({𝑋} ∪ 𝑆)
5553, 54syl6ss 3983 . . . . . . 7 (𝑗𝑆𝑗 ⊆ ({𝑋} ∪ 𝑆))
56 ssfii 8872 . . . . . . . 8 (({𝑋} ∪ 𝑆) ∈ V → ({𝑋} ∪ 𝑆) ⊆ (fi‘({𝑋} ∪ 𝑆)))
5747, 56syl 17 . . . . . . 7 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → ({𝑋} ∪ 𝑆) ⊆ (fi‘({𝑋} ∪ 𝑆)))
5855, 57sylan9ssr 3985 . . . . . 6 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑗𝑆) → 𝑗 ⊆ (fi‘({𝑋} ∪ 𝑆)))
59 bastg 21490 . . . . . . 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 3983 . . . . 5 (((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑗𝑆) → 𝑗 ⊆ (topGen‘(fi‘({𝑋} ∪ 𝑆))))
6261ralrimiva 3187 . . . 4 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → ∀𝑗𝑆 𝑗 ⊆ (topGen‘(fi‘({𝑋} ∪ 𝑆))))
63 sseq2 3997 . . . . . 6 (𝑘 = (topGen‘(fi‘({𝑋} ∪ 𝑆))) → (𝑗𝑘𝑗 ⊆ (topGen‘(fi‘({𝑋} ∪ 𝑆)))))
6463ralbidv 3202 . . . . 5 (𝑘 = (topGen‘(fi‘({𝑋} ∪ 𝑆))) → (∀𝑗𝑆 𝑗𝑘 ↔ ∀𝑗𝑆 𝑗 ⊆ (topGen‘(fi‘({𝑋} ∪ 𝑆)))))
6564elrab 3684 . . . 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 583 . . 3 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → (topGen‘(fi‘({𝑋} ∪ 𝑆))) ∈ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑗𝑘})
67 intss1 4889 . . 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 3988 1 ((𝑋𝑉𝑆 ⊆ (TopOn‘𝑋)) → (topGen‘(fi‘({𝑋} ∪ 𝑆))) = {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗𝑆 𝑗𝑘})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  wral 3143  {crab 3147  Vcvv 3500  cun 3938  wss 3940  𝒫 cpw 4542  {csn 4564   cuni 4837   cint 4874  cfv 6352  ficfi 8863  topGenctg 16701  Topctop 21417  TopOnctopon 21434  TopBasesctb 21469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7569  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-oadd 8097  df-er 8279  df-en 8499  df-fin 8502  df-fi 8864  df-topgen 16707  df-top 21418  df-topon 21435  df-bases 21470
This theorem is referenced by: (None)
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