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Theorem tposeq 6112
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 3121 . . 3  |-  ( F  =  G  ->  F  C_  G )
2 tposss 6111 . . 3  |-  ( F 
C_  G  -> tpos  F  C_ tpos  G )
31, 2syl 14 . 2  |-  ( F  =  G  -> tpos  F  C_ tpos  G )
4 eqimss2 3122 . . 3  |-  ( F  =  G  ->  G  C_  F )
5 tposss 6111 . . 3  |-  ( G 
C_  F  -> tpos  G  C_ tpos  F )
64, 5syl 14 . 2  |-  ( F  =  G  -> tpos  G  C_ tpos  F )
73, 6eqssd 3084 1  |-  ( F  =  G  -> tpos  F  = tpos 
G )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1316    C_ wss 3041  tpos ctpos 6109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-mpt 3961  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-res 4521  df-tpos 6110
This theorem is referenced by:  tposeqd  6113  tposeqi  6142
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