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Theorem tposfo 6784
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 4959 . . 3  |-  Rel  ( A  X.  B )
2 tposfo2 6780 . . 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 5267 . . 3  |-  `' ( A  X.  B )  =  ( B  X.  A )
5 foeq2 5629 . . 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 1369    X. cxp 4850   `'ccnv 4851   Rel wrel 4857   -onto->wfo 5428  tpos ctpos 6756
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-sbc 3199  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-fo 5436  df-fv 5438  df-tpos 6757
This theorem is referenced by: (None)
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