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Theorem tgtop11 19990
Description: The topology generation function is one-to-one when applied to completed topologies. (Contributed by NM, 18-Jul-2006.)
Assertion
Ref Expression
tgtop11  |-  ( ( J  e.  Top  /\  K  e.  Top  /\  ( topGen `
 J )  =  ( topGen `  K )
)  ->  J  =  K )

Proof of Theorem tgtop11
StepHypRef Expression
1 tgtop 19981 . . 3  |-  ( J  e.  Top  ->  ( topGen `
 J )  =  J )
2 tgtop 19981 . . 3  |-  ( K  e.  Top  ->  ( topGen `
 K )  =  K )
31, 2eqeqan12d 2446 . 2  |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  ( ( topGen `  J
)  =  ( topGen `  K )  <->  J  =  K ) )
43biimp3a 1365 1  |-  ( ( J  e.  Top  /\  K  e.  Top  /\  ( topGen `
 J )  =  ( topGen `  K )
)  ->  J  =  K )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 983    = wceq 1438    e. wcel 1869   ` cfv 5599   topGenctg 15329   Topctop 19909
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-sbc 3301  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-br 4422  df-opab 4481  df-mpt 4482  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-iota 5563  df-fun 5601  df-fv 5607  df-topgen 15335  df-top 19913
This theorem is referenced by: (None)
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