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Theorem tposfo 7000
Description: The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.)
Assertion
Ref Expression
tposfo  |-  ( F : ( A  X.  B ) -onto-> C  -> tpos  F : ( B  X.  A ) -onto-> C )

Proof of Theorem tposfo
StepHypRef Expression
1 relxp 5119 . . 3  |-  Rel  ( A  X.  B )
2 tposfo2 6996 . . 3  |-  ( Rel  ( A  X.  B
)  ->  ( F : ( A  X.  B ) -onto-> C  -> tpos  F : `' ( A  X.  B ) -onto-> C ) )
31, 2ax-mp 5 . 2  |-  ( F : ( A  X.  B ) -onto-> C  -> tpos  F : `' ( A  X.  B ) -onto-> C )
4 cnvxp 5431 . . 3  |-  `' ( A  X.  B )  =  ( B  X.  A )
5 foeq2 5798 . . 3  |-  ( `' ( A  X.  B
)  =  ( B  X.  A )  -> 
(tpos  F : `' ( A  X.  B
) -onto-> C  <-> tpos  F : ( B  X.  A ) -onto-> C ) )
64, 5ax-mp 5 . 2  |-  (tpos  F : `' ( A  X.  B ) -onto-> C  <-> tpos  F : ( B  X.  A )
-onto-> C )
73, 6sylib 196 1  |-  ( F : ( A  X.  B ) -onto-> C  -> tpos  F : ( B  X.  A ) -onto-> C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1395    X. cxp 5006   `'ccnv 5007   Rel wrel 5013   -onto->wfo 5592  tpos ctpos 6972
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-fo 5600  df-fv 5602  df-tpos 6973
This theorem is referenced by: (None)
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