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Theorem tposeq 6473
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 3392 . . 3  |-  ( F  =  G  ->  F  C_  G )
2 tposss 6472 . . 3  |-  ( F 
C_  G  -> tpos  F  C_ tpos  G )
31, 2syl 16 . 2  |-  ( F  =  G  -> tpos  F  C_ tpos  G )
4 eqimss2 3393 . . 3  |-  ( F  =  G  ->  G  C_  F )
5 tposss 6472 . . 3  |-  ( G 
C_  F  -> tpos  G  C_ tpos  F )
64, 5syl 16 . 2  |-  ( F  =  G  -> tpos  G  C_ tpos  F )
73, 6eqssd 3357 1  |-  ( F  =  G  -> tpos  F  = tpos 
G )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    C_ wss 3312  tpos ctpos 6470
This theorem is referenced by:  tposeqd  6474  tposeqi  6504
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-mpt 4260  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-res 4882  df-tpos 6471
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