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Theorem f1ocan2fv 31757
Description: Cancel a composition by the converse of a bijection by preapplying the bijection. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
f1ocan2fv  |-  ( ( Fun  F  /\  G : A -1-1-onto-> B  /\  X  e.  A )  ->  (
( F  o.  `' G ) `  ( G `  X )
)  =  ( F `
 X ) )

Proof of Theorem f1ocan2fv
StepHypRef Expression
1 f1orel 5834 . . . . . 6  |-  ( G : A -1-1-onto-> B  ->  Rel  G )
2 dfrel2 5306 . . . . . 6  |-  ( Rel 
G  <->  `' `' G  =  G
)
31, 2sylib 199 . . . . 5  |-  ( G : A -1-1-onto-> B  ->  `' `' G  =  G )
433ad2ant2 1027 . . . 4  |-  ( ( Fun  F  /\  G : A -1-1-onto-> B  /\  X  e.  A )  ->  `' `' G  =  G
)
54fveq1d 5883 . . 3  |-  ( ( Fun  F  /\  G : A -1-1-onto-> B  /\  X  e.  A )  ->  ( `' `' G `  X )  =  ( G `  X ) )
65fveq2d 5885 . 2  |-  ( ( Fun  F  /\  G : A -1-1-onto-> B  /\  X  e.  A )  ->  (
( F  o.  `' G ) `  ( `' `' G `  X ) )  =  ( ( F  o.  `' G
) `  ( G `  X ) ) )
7 f1ocnv 5843 . . 3  |-  ( G : A -1-1-onto-> B  ->  `' G : B -1-1-onto-> A )
8 f1ocan1fv 31756 . . 3  |-  ( ( Fun  F  /\  `' G : B -1-1-onto-> A  /\  X  e.  A )  ->  (
( F  o.  `' G ) `  ( `' `' G `  X ) )  =  ( F `
 X ) )
97, 8syl3an2 1298 . 2  |-  ( ( Fun  F  /\  G : A -1-1-onto-> B  /\  X  e.  A )  ->  (
( F  o.  `' G ) `  ( `' `' G `  X ) )  =  ( F `
 X ) )
106, 9eqtr3d 2472 1  |-  ( ( Fun  F  /\  G : A -1-1-onto-> B  /\  X  e.  A )  ->  (
( F  o.  `' G ) `  ( G `  X )
)  =  ( F `
 X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 982    = wceq 1437    e. wcel 1870   `'ccnv 4853    o. ccom 4858   Rel wrel 4859   Fun wfun 5595   -1-1-onto->wf1o 5600   ` cfv 5601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609
This theorem is referenced by: (None)
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