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Theorem tposeq 5893
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 3025 . . 3  |-  ( F  =  G  ->  F  C_  G )
2 tposss 5892 . . 3  |-  ( F 
C_  G  -> tpos  F  C_ tpos  G )
31, 2syl 14 . 2  |-  ( F  =  G  -> tpos  F  C_ tpos  G )
4 eqimss2 3026 . . 3  |-  ( F  =  G  ->  G  C_  F )
5 tposss 5892 . . 3  |-  ( G 
C_  F  -> tpos  G  C_ tpos  F )
64, 5syl 14 . 2  |-  ( F  =  G  -> tpos  G  C_ tpos  F )
73, 6eqssd 2990 1  |-  ( F  =  G  -> tpos  F  = tpos 
G )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1259    C_ wss 2945  tpos ctpos 5890
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-mpt 3848  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-res 4385  df-tpos 5891
This theorem is referenced by:  tposeqd  5894  tposeqi  5923
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