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Theorem tposeq 6236
Description: Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
Assertion
Ref Expression
tposeq  |-  ( F  =  G  -> tpos  F  = tpos 
G )

Proof of Theorem tposeq
StepHypRef Expression
1 eqimss 3230 . . 3  |-  ( F  =  G  ->  F  C_  G )
2 tposss 6235 . . 3  |-  ( F 
C_  G  -> tpos  F  C_ tpos  G )
31, 2syl 15 . 2  |-  ( F  =  G  -> tpos  F  C_ tpos  G )
4 eqimss2 3231 . . 3  |-  ( F  =  G  ->  G  C_  F )
5 tposss 6235 . . 3  |-  ( G 
C_  F  -> tpos  G  C_ tpos  F )
64, 5syl 15 . 2  |-  ( F  =  G  -> tpos  G  C_ tpos  F )
73, 6eqssd 3196 1  |-  ( F  =  G  -> tpos  F  = tpos 
G )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    C_ wss 3152  tpos ctpos 6233
This theorem is referenced by:  tposeqd  6237  tposeqi  6267  oppcval  13616  oppchomfval  13617  oppccofval  13619  oppcmon  13641  oppgval  14820  oppgplusfval  14821  oppglsm  14953  opprval  15406  opprmulfval  15407
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
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-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-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-mpt 4079  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-res 4701  df-tpos 6234
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