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Theorem f1ocan2fv 26429
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 5677 . . . . . 6  |-  ( G : A -1-1-onto-> B  ->  Rel  G )
2 dfrel2 5321 . . . . . 6  |-  ( Rel 
G  <->  `' `' G  =  G
)
31, 2sylib 189 . . . . 5  |-  ( G : A -1-1-onto-> B  ->  `' `' G  =  G )
433ad2ant2 979 . . . 4  |-  ( ( Fun  F  /\  G : A -1-1-onto-> B  /\  X  e.  A )  ->  `' `' G  =  G
)
54fveq1d 5730 . . 3  |-  ( ( Fun  F  /\  G : A -1-1-onto-> B  /\  X  e.  A )  ->  ( `' `' G `  X )  =  ( G `  X ) )
65fveq2d 5732 . 2  |-  ( ( Fun  F  /\  G : A -1-1-onto-> B  /\  X  e.  A )  ->  (
( F  o.  `' G ) `  ( `' `' G `  X ) )  =  ( ( F  o.  `' G
) `  ( G `  X ) ) )
7 f1ocnv 5687 . . 3  |-  ( G : A -1-1-onto-> B  ->  `' G : B -1-1-onto-> A )
8 f1ocan1fv 26428 . . 3  |-  ( ( Fun  F  /\  `' G : B -1-1-onto-> A  /\  X  e.  A )  ->  (
( F  o.  `' G ) `  ( `' `' G `  X ) )  =  ( F `
 X ) )
97, 8syl3an2 1218 . 2  |-  ( ( Fun  F  /\  G : A -1-1-onto-> B  /\  X  e.  A )  ->  (
( F  o.  `' G ) `  ( `' `' G `  X ) )  =  ( F `
 X ) )
106, 9eqtr3d 2470 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 936    = wceq 1652    e. wcel 1725   `'ccnv 4877    o. ccom 4882   Rel wrel 4883   Fun wfun 5448   -1-1-onto->wf1o 5453   ` cfv 5454
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462
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