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Theorem eltg2 13556
Description: Membership in a topology generated by a basis. (Contributed by NM, 15-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
Assertion
Ref Expression
eltg2 (𝐡 ∈ 𝑉 β†’ (𝐴 ∈ (topGenβ€˜π΅) ↔ (𝐴 βŠ† βˆͺ 𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝐴))))
Distinct variable groups:   π‘₯,𝑦,𝐴   π‘₯,𝐡,𝑦   π‘₯,𝑉,𝑦
Proof of Theorem eltg2
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 tgval2 13554 . . 3 (𝐡 ∈ 𝑉 β†’ (topGenβ€˜π΅) = {𝑧 ∣ (𝑧 βŠ† βˆͺ 𝐡 ∧ βˆ€π‘₯ ∈ 𝑧 βˆƒπ‘¦ ∈ 𝐡 (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝑧))})
21eleq2d 2247 . 2 (𝐡 ∈ 𝑉 β†’ (𝐴 ∈ (topGenβ€˜π΅) ↔ 𝐴 ∈ {𝑧 ∣ (𝑧 βŠ† βˆͺ 𝐡 ∧ βˆ€π‘₯ ∈ 𝑧 βˆƒπ‘¦ ∈ 𝐡 (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝑧))}))
3 elex 2749 . . . 4 (𝐴 ∈ {𝑧 ∣ (𝑧 βŠ† βˆͺ 𝐡 ∧ βˆ€π‘₯ ∈ 𝑧 βˆƒπ‘¦ ∈ 𝐡 (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝑧))} β†’ 𝐴 ∈ V)
43adantl 277 . . 3 ((𝐡 ∈ 𝑉 ∧ 𝐴 ∈ {𝑧 ∣ (𝑧 βŠ† βˆͺ 𝐡 ∧ βˆ€π‘₯ ∈ 𝑧 βˆƒπ‘¦ ∈ 𝐡 (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝑧))}) β†’ 𝐴 ∈ V)
5 uniexg 4440 . . . . . 6 (𝐡 ∈ 𝑉 β†’ βˆͺ 𝐡 ∈ V)
6 ssexg 4143 . . . . . 6 ((𝐴 βŠ† βˆͺ 𝐡 ∧ βˆͺ 𝐡 ∈ V) β†’ 𝐴 ∈ V)
75, 6sylan2 286 . . . . 5 ((𝐴 βŠ† βˆͺ 𝐡 ∧ 𝐡 ∈ 𝑉) β†’ 𝐴 ∈ V)
87ancoms 268 . . . 4 ((𝐡 ∈ 𝑉 ∧ 𝐴 βŠ† βˆͺ 𝐡) β†’ 𝐴 ∈ V)
98adantrr 479 . . 3 ((𝐡 ∈ 𝑉 ∧ (𝐴 βŠ† βˆͺ 𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝐴))) β†’ 𝐴 ∈ V)
10 sseq1 3179 . . . . 5 (𝑧 = 𝐴 β†’ (𝑧 βŠ† βˆͺ 𝐡 ↔ 𝐴 βŠ† βˆͺ 𝐡))
11 sseq2 3180 . . . . . . . 8 (𝑧 = 𝐴 β†’ (𝑦 βŠ† 𝑧 ↔ 𝑦 βŠ† 𝐴))
1211anbi2d 464 . . . . . . 7 (𝑧 = 𝐴 β†’ ((π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝑧) ↔ (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝐴)))
1312rexbidv 2478 . . . . . 6 (𝑧 = 𝐴 β†’ (βˆƒπ‘¦ ∈ 𝐡 (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝑧) ↔ βˆƒπ‘¦ ∈ 𝐡 (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝐴)))
1413raleqbi1dv 2681 . . . . 5 (𝑧 = 𝐴 β†’ (βˆ€π‘₯ ∈ 𝑧 βˆƒπ‘¦ ∈ 𝐡 (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝑧) ↔ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝐴)))
1510, 14anbi12d 473 . . . 4 (𝑧 = 𝐴 β†’ ((𝑧 βŠ† βˆͺ 𝐡 ∧ βˆ€π‘₯ ∈ 𝑧 βˆƒπ‘¦ ∈ 𝐡 (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝑧)) ↔ (𝐴 βŠ† βˆͺ 𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝐴))))
1615elabg 2884 . . 3 (𝐴 ∈ V β†’ (𝐴 ∈ {𝑧 ∣ (𝑧 βŠ† βˆͺ 𝐡 ∧ βˆ€π‘₯ ∈ 𝑧 βˆƒπ‘¦ ∈ 𝐡 (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝑧))} ↔ (𝐴 βŠ† βˆͺ 𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝐴))))
174, 9, 16pm5.21nd 916 . 2 (𝐡 ∈ 𝑉 β†’ (𝐴 ∈ {𝑧 ∣ (𝑧 βŠ† βˆͺ 𝐡 ∧ βˆ€π‘₯ ∈ 𝑧 βˆƒπ‘¦ ∈ 𝐡 (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝑧))} ↔ (𝐴 βŠ† βˆͺ 𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝐴))))
182, 17bitrd 188 1 (𝐡 ∈ 𝑉 β†’ (𝐴 ∈ (topGenβ€˜π΅) ↔ (𝐴 βŠ† βˆͺ 𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† 𝐴))))
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353   ∈ wcel 2148  {cab 2163  βˆ€wral 2455  βˆƒwrex 2456  Vcvv 2738   βŠ† wss 3130  βˆͺ cuni 3810  β€˜cfv 5217  topGenctg 12703
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-sbc 2964  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-iota 5179  df-fun 5219  df-fv 5225  df-topgen 12709
This theorem is referenced by:  eltg2b  13557  tg1  13562  tgcl  13567  elmopn  13949  xmettx  14013
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