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Theorem tposeq 6417
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 3343 . . 3  |-  ( F  =  G  ->  F  C_  G )
2 tposss 6416 . . 3  |-  ( F 
C_  G  -> tpos  F  C_ tpos  G )
31, 2syl 16 . 2  |-  ( F  =  G  -> tpos  F  C_ tpos  G )
4 eqimss2 3344 . . 3  |-  ( F  =  G  ->  G  C_  F )
5 tposss 6416 . . 3  |-  ( G 
C_  F  -> tpos  G  C_ tpos  F )
64, 5syl 16 . 2  |-  ( F  =  G  -> tpos  G  C_ tpos  F )
73, 6eqssd 3308 1  |-  ( F  =  G  -> tpos  F  = tpos 
G )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    C_ wss 3263  tpos ctpos 6414
This theorem is referenced by:  tposeqd  6418  tposeqi  6448
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-br 4154  df-opab 4208  df-mpt 4209  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-res 4830  df-tpos 6415
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