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Theorem bastop1 13622
Description: A subset of a topology is a basis for the topology iff every member of the topology is a union of members of the basis. We use the idiom "(topGenβ€˜π΅) = 𝐽 " to express "𝐡 is a basis for topology 𝐽 " since we do not have a separate notation for this. Definition 15.35 of [Schechter] p. 428. (Contributed by NM, 2-Feb-2008.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)
Assertion
Ref Expression
bastop1 ((𝐽 ∈ Top ∧ 𝐡 βŠ† 𝐽) β†’ ((topGenβ€˜π΅) = 𝐽 ↔ βˆ€π‘₯ ∈ 𝐽 βˆƒπ‘¦(𝑦 βŠ† 𝐡 ∧ π‘₯ = βˆͺ 𝑦)))
Distinct variable groups:   π‘₯,𝑦,𝐡   π‘₯,𝐽,𝑦
Proof of Theorem bastop1
StepHypRef Expression
1 tgss 13602 . . . . 5 ((𝐽 ∈ Top ∧ 𝐡 βŠ† 𝐽) β†’ (topGenβ€˜π΅) βŠ† (topGenβ€˜π½))
2 tgtop 13607 . . . . . 6 (𝐽 ∈ Top β†’ (topGenβ€˜π½) = 𝐽)
32adantr 276 . . . . 5 ((𝐽 ∈ Top ∧ 𝐡 βŠ† 𝐽) β†’ (topGenβ€˜π½) = 𝐽)
41, 3sseqtrd 3195 . . . 4 ((𝐽 ∈ Top ∧ 𝐡 βŠ† 𝐽) β†’ (topGenβ€˜π΅) βŠ† 𝐽)
5 eqss 3172 . . . . 5 ((topGenβ€˜π΅) = 𝐽 ↔ ((topGenβ€˜π΅) βŠ† 𝐽 ∧ 𝐽 βŠ† (topGenβ€˜π΅)))
65baib 919 . . . 4 ((topGenβ€˜π΅) βŠ† 𝐽 β†’ ((topGenβ€˜π΅) = 𝐽 ↔ 𝐽 βŠ† (topGenβ€˜π΅)))
74, 6syl 14 . . 3 ((𝐽 ∈ Top ∧ 𝐡 βŠ† 𝐽) β†’ ((topGenβ€˜π΅) = 𝐽 ↔ 𝐽 βŠ† (topGenβ€˜π΅)))
8 dfss3 3147 . . 3 (𝐽 βŠ† (topGenβ€˜π΅) ↔ βˆ€π‘₯ ∈ 𝐽 π‘₯ ∈ (topGenβ€˜π΅))
97, 8bitrdi 196 . 2 ((𝐽 ∈ Top ∧ 𝐡 βŠ† 𝐽) β†’ ((topGenβ€˜π΅) = 𝐽 ↔ βˆ€π‘₯ ∈ 𝐽 π‘₯ ∈ (topGenβ€˜π΅)))
10 ssexg 4144 . . . . 5 ((𝐡 βŠ† 𝐽 ∧ 𝐽 ∈ Top) β†’ 𝐡 ∈ V)
1110ancoms 268 . . . 4 ((𝐽 ∈ Top ∧ 𝐡 βŠ† 𝐽) β†’ 𝐡 ∈ V)
12 eltg3 13596 . . . 4 (𝐡 ∈ V β†’ (π‘₯ ∈ (topGenβ€˜π΅) ↔ βˆƒπ‘¦(𝑦 βŠ† 𝐡 ∧ π‘₯ = βˆͺ 𝑦)))
1311, 12syl 14 . . 3 ((𝐽 ∈ Top ∧ 𝐡 βŠ† 𝐽) β†’ (π‘₯ ∈ (topGenβ€˜π΅) ↔ βˆƒπ‘¦(𝑦 βŠ† 𝐡 ∧ π‘₯ = βˆͺ 𝑦)))
1413ralbidv 2477 . 2 ((𝐽 ∈ Top ∧ 𝐡 βŠ† 𝐽) β†’ (βˆ€π‘₯ ∈ 𝐽 π‘₯ ∈ (topGenβ€˜π΅) ↔ βˆ€π‘₯ ∈ 𝐽 βˆƒπ‘¦(𝑦 βŠ† 𝐡 ∧ π‘₯ = βˆͺ 𝑦)))
159, 14bitrd 188 1 ((𝐽 ∈ Top ∧ 𝐡 βŠ† 𝐽) β†’ ((topGenβ€˜π΅) = 𝐽 ↔ βˆ€π‘₯ ∈ 𝐽 βˆƒπ‘¦(𝑦 βŠ† 𝐡 ∧ π‘₯ = βˆͺ 𝑦)))
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353  βˆƒwex 1492   ∈ wcel 2148  βˆ€wral 2455  Vcvv 2739   βŠ† wss 3131  βˆͺ cuni 3811  β€˜cfv 5218  topGenctg 12708  Topctop 13536
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 4123  ax-pow 4176  ax-pr 4211  ax-un 4435
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 2741  df-sbc 2965  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-topgen 12714  df-top 13537
This theorem is referenced by:  bastop2  13623
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