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Theorem tposeq 6245
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 3209 . . 3  |-  ( F  =  G  ->  F  C_  G )
2 tposss 6244 . . 3  |-  ( F 
C_  G  -> tpos  F  C_ tpos  G )
31, 2syl 14 . 2  |-  ( F  =  G  -> tpos  F  C_ tpos  G )
4 eqimss2 3210 . . 3  |-  ( F  =  G  ->  G  C_  F )
5 tposss 6244 . . 3  |-  ( G 
C_  F  -> tpos  G  C_ tpos  F )
64, 5syl 14 . 2  |-  ( F  =  G  -> tpos  G  C_ tpos  F )
73, 6eqssd 3172 1  |-  ( F  =  G  -> tpos  F  = tpos 
G )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353    C_ wss 3129  tpos ctpos 6242
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4003  df-opab 4064  df-mpt 4065  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-res 4637  df-tpos 6243
This theorem is referenced by:  tposeqd  6246  tposeqi  6275
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