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Theorem tposeq 6238
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 3207 . . 3  |-  ( F  =  G  ->  F  C_  G )
2 tposss 6237 . . 3  |-  ( F 
C_  G  -> tpos  F  C_ tpos  G )
31, 2syl 14 . 2  |-  ( F  =  G  -> tpos  F  C_ tpos  G )
4 eqimss2 3208 . . 3  |-  ( F  =  G  ->  G  C_  F )
5 tposss 6237 . . 3  |-  ( G 
C_  F  -> tpos  G  C_ tpos  F )
64, 5syl 14 . 2  |-  ( F  =  G  -> tpos  G  C_ tpos  F )
73, 6eqssd 3170 1  |-  ( F  =  G  -> tpos  F  = tpos 
G )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353    C_ wss 3127  tpos ctpos 6235
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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-br 3999  df-opab 4060  df-mpt 4061  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-res 4632  df-tpos 6236
This theorem is referenced by:  tposeqd  6239  tposeqi  6268
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