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Theorem f1ocan2fv 26498
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 5491 . . . . . 6  |-  ( G : A -1-1-onto-> B  ->  Rel  G )
2 dfrel2 5140 . . . . . 6  |-  ( Rel 
G  <->  `' `' G  =  G
)
31, 2sylib 188 . . . . 5  |-  ( G : A -1-1-onto-> B  ->  `' `' G  =  G )
433ad2ant2 977 . . . 4  |-  ( ( Fun  F  /\  G : A -1-1-onto-> B  /\  X  e.  A )  ->  `' `' G  =  G
)
54fveq1d 5543 . . 3  |-  ( ( Fun  F  /\  G : A -1-1-onto-> B  /\  X  e.  A )  ->  ( `' `' G `  X )  =  ( G `  X ) )
65fveq2d 5545 . 2  |-  ( ( Fun  F  /\  G : A -1-1-onto-> B  /\  X  e.  A )  ->  (
( F  o.  `' G ) `  ( `' `' G `  X ) )  =  ( ( F  o.  `' G
) `  ( G `  X ) ) )
7 f1ocnv 5501 . . 3  |-  ( G : A -1-1-onto-> B  ->  `' G : B -1-1-onto-> A )
8 f1ocan1fv 26497 . . 3  |-  ( ( Fun  F  /\  `' G : B -1-1-onto-> A  /\  X  e.  A )  ->  (
( F  o.  `' G ) `  ( `' `' G `  X ) )  =  ( F `
 X ) )
97, 8syl3an2 1216 . 2  |-  ( ( Fun  F  /\  G : A -1-1-onto-> B  /\  X  e.  A )  ->  (
( F  o.  `' G ) `  ( `' `' G `  X ) )  =  ( F `
 X ) )
106, 9eqtr3d 2330 1  |-  ( ( Fun  F  /\  G : A -1-1-onto-> B  /\  X  e.  A )  ->  (
( F  o.  `' G ) `  ( G `  X )
)  =  ( F `
 X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1632    e. wcel 1696   `'ccnv 4704    o. ccom 4709   Rel wrel 4710   Fun wfun 5265   -1-1-onto->wf1o 5270   ` cfv 5271
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279
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