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Theorem tposf 6349
Description: The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.)
Assertion
Ref Expression
tposf  |-  ( F : ( A  X.  B ) --> C  -> tpos  F : ( B  X.  A ) --> C )

Proof of Theorem tposf
StepHypRef Expression
1 relxp 4876 . . 3  |-  Rel  ( A  X.  B )
2 tposf2 6345 . . 3  |-  ( Rel  ( A  X.  B
)  ->  ( F : ( A  X.  B ) --> C  -> tpos  F : `' ( A  X.  B ) --> C ) )
31, 2ax-mp 8 . 2  |-  ( F : ( A  X.  B ) --> C  -> tpos  F : `' ( A  X.  B ) --> C )
4 cnvxp 5179 . . 3  |-  `' ( A  X.  B )  =  ( B  X.  A )
54feq2i 5467 . 2  |-  (tpos  F : `' ( A  X.  B ) --> C  <-> tpos  F : ( B  X.  A ) --> C )
63, 5sylib 188 1  |-  ( F : ( A  X.  B ) --> C  -> tpos  F : ( B  X.  A ) --> C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    X. cxp 4769   `'ccnv 4770   Rel wrel 4776   -->wf 5333  tpos ctpos 6320
This theorem is referenced by:  tposfn  6350
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-fo 5343  df-fv 5345  df-tpos 6321
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