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Theorem basgen 16726
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 16706 . . . 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 16711 . . . 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 3214 . 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 3196 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 1623    e. wcel 1684    C_ wss 3152   ` cfv 5255   topGenctg 13342   Topctop 16631
This theorem is referenced by:  basgen2  16727
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-topgen 13344  df-top 16636
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