MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tposf Structured version   Unicode version

Theorem tposf 6984
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 5110 . . 3  |-  Rel  ( A  X.  B )
2 tposf2 6980 . . 3  |-  ( Rel  ( A  X.  B
)  ->  ( F : ( A  X.  B ) --> C  -> tpos  F : `' ( A  X.  B ) --> C ) )
31, 2ax-mp 5 . 2  |-  ( F : ( A  X.  B ) --> C  -> tpos  F : `' ( A  X.  B ) --> C )
4 cnvxp 5424 . . 3  |-  `' ( A  X.  B )  =  ( B  X.  A )
54feq2i 5724 . 2  |-  (tpos  F : `' ( A  X.  B ) --> C  <-> tpos  F : ( B  X.  A ) --> C )
63, 5sylib 196 1  |-  ( F : ( A  X.  B ) --> C  -> tpos  F : ( B  X.  A ) --> C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    X. cxp 4997   `'ccnv 4998   Rel wrel 5004   -->wf 5584  tpos ctpos 6955
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fo 5594  df-fv 5596  df-tpos 6956
This theorem is referenced by:  tposfn  6985  mattposcl  18762  tposmap  18766
  Copyright terms: Public domain W3C validator