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Theorem basgen 16742
Description: Given a topology  J, show that a subset  B satisfying the third antecedent is a basis for it. Lemma 2.3 of [Munkres] p. 81 using abbreviations. (Contributed by NM, 22-Jul-2006.) (Revised by Mario Carneiro, 2-Sep-2015.)
Assertion
Ref Expression
basgen  |-  ( ( J  e.  Top  /\  B  C_  J  /\  J  C_  ( topGen `  B )
)  ->  ( topGen `  B )  =  J )

Proof of Theorem basgen
StepHypRef Expression
1 tgss 16722 . . . 4  |-  ( ( J  e.  Top  /\  B  C_  J )  -> 
( topGen `  B )  C_  ( topGen `  J )
)
213adant3 975 . . 3  |-  ( ( J  e.  Top  /\  B  C_  J  /\  J  C_  ( topGen `  B )
)  ->  ( topGen `  B )  C_  ( topGen `
 J ) )
3 tgtop 16727 . . . 4  |-  ( J  e.  Top  ->  ( topGen `
 J )  =  J )
433ad2ant1 976 . . 3  |-  ( ( J  e.  Top  /\  B  C_  J  /\  J  C_  ( topGen `  B )
)  ->  ( topGen `  J )  =  J )
52, 4sseqtrd 3227 . 2  |-  ( ( J  e.  Top  /\  B  C_  J  /\  J  C_  ( topGen `  B )
)  ->  ( topGen `  B )  C_  J
)
6 simp3 957 . 2  |-  ( ( J  e.  Top  /\  B  C_  J  /\  J  C_  ( topGen `  B )
)  ->  J  C_  ( topGen `
 B ) )
75, 6eqssd 3209 1  |-  ( ( J  e.  Top  /\  B  C_  J  /\  J  C_  ( topGen `  B )
)  ->  ( topGen `  B )  =  J )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1632    e. wcel 1696    C_ wss 3165   ` cfv 5271   topGenctg 13358   Topctop 16647
This theorem is referenced by:  basgen2  16743
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-topgen 13360  df-top 16652
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