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Theorem f1ocan2fv 29849
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 5819 . . . . . 6  |-  ( G : A -1-1-onto-> B  ->  Rel  G )
2 dfrel2 5457 . . . . . 6  |-  ( Rel 
G  <->  `' `' G  =  G
)
31, 2sylib 196 . . . . 5  |-  ( G : A -1-1-onto-> B  ->  `' `' G  =  G )
433ad2ant2 1018 . . . 4  |-  ( ( Fun  F  /\  G : A -1-1-onto-> B  /\  X  e.  A )  ->  `' `' G  =  G
)
54fveq1d 5868 . . 3  |-  ( ( Fun  F  /\  G : A -1-1-onto-> B  /\  X  e.  A )  ->  ( `' `' G `  X )  =  ( G `  X ) )
65fveq2d 5870 . 2  |-  ( ( Fun  F  /\  G : A -1-1-onto-> B  /\  X  e.  A )  ->  (
( F  o.  `' G ) `  ( `' `' G `  X ) )  =  ( ( F  o.  `' G
) `  ( G `  X ) ) )
7 f1ocnv 5828 . . 3  |-  ( G : A -1-1-onto-> B  ->  `' G : B -1-1-onto-> A )
8 f1ocan1fv 29848 . . 3  |-  ( ( Fun  F  /\  `' G : B -1-1-onto-> A  /\  X  e.  A )  ->  (
( F  o.  `' G ) `  ( `' `' G `  X ) )  =  ( F `
 X ) )
97, 8syl3an2 1262 . 2  |-  ( ( Fun  F  /\  G : A -1-1-onto-> B  /\  X  e.  A )  ->  (
( F  o.  `' G ) `  ( `' `' G `  X ) )  =  ( F `
 X ) )
106, 9eqtr3d 2510 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 973    = wceq 1379    e. wcel 1767   `'ccnv 4998    o. ccom 5003   Rel wrel 5004   Fun wfun 5582   -1-1-onto->wf1o 5587   ` cfv 5588
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
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-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  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-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596
This theorem is referenced by: (None)
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