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Theorem f1ocan1fv 26419
Description: Cancel a composition by a bijection by preapplying the converse. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)
Assertion
Ref Expression
f1ocan1fv  |-  ( ( Fun  F  /\  G : A -1-1-onto-> B  /\  X  e.  B )  ->  (
( F  o.  G
) `  ( `' G `  X )
)  =  ( F `
 X ) )

Proof of Theorem f1ocan1fv
StepHypRef Expression
1 f1of 5666 . . . 4  |-  ( G : A -1-1-onto-> B  ->  G : A
--> B )
213ad2ant2 979 . . 3  |-  ( ( Fun  F  /\  G : A -1-1-onto-> B  /\  X  e.  B )  ->  G : A --> B )
3 f1ocnv 5679 . . . . . 6  |-  ( G : A -1-1-onto-> B  ->  `' G : B -1-1-onto-> A )
4 f1of 5666 . . . . . 6  |-  ( `' G : B -1-1-onto-> A  ->  `' G : B --> A )
53, 4syl 16 . . . . 5  |-  ( G : A -1-1-onto-> B  ->  `' G : B --> A )
653ad2ant2 979 . . . 4  |-  ( ( Fun  F  /\  G : A -1-1-onto-> B  /\  X  e.  B )  ->  `' G : B --> A )
7 simp3 959 . . . 4  |-  ( ( Fun  F  /\  G : A -1-1-onto-> B  /\  X  e.  B )  ->  X  e.  B )
86, 7ffvelrnd 5863 . . 3  |-  ( ( Fun  F  /\  G : A -1-1-onto-> B  /\  X  e.  B )  ->  ( `' G `  X )  e.  A )
9 fvco3 5792 . . 3  |-  ( ( G : A --> B  /\  ( `' G `  X )  e.  A )  -> 
( ( F  o.  G ) `  ( `' G `  X ) )  =  ( F `
 ( G `  ( `' G `  X ) ) ) )
102, 8, 9syl2anc 643 . 2  |-  ( ( Fun  F  /\  G : A -1-1-onto-> B  /\  X  e.  B )  ->  (
( F  o.  G
) `  ( `' G `  X )
)  =  ( F `
 ( G `  ( `' G `  X ) ) ) )
11 f1ocnvfv2 6007 . . . 4  |-  ( ( G : A -1-1-onto-> B  /\  X  e.  B )  ->  ( G `  ( `' G `  X ) )  =  X )
12113adant1 975 . . 3  |-  ( ( Fun  F  /\  G : A -1-1-onto-> B  /\  X  e.  B )  ->  ( G `  ( `' G `  X )
)  =  X )
1312fveq2d 5724 . 2  |-  ( ( Fun  F  /\  G : A -1-1-onto-> B  /\  X  e.  B )  ->  ( F `  ( G `  ( `' G `  X ) ) )  =  ( F `  X ) )
1410, 13eqtrd 2467 1  |-  ( ( Fun  F  /\  G : A -1-1-onto-> B  /\  X  e.  B )  ->  (
( 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 4869    o. ccom 4874   Fun wfun 5440   -->wf 5442   -1-1-onto->wf1o 5445   ` cfv 5446
This theorem is referenced by:  f1ocan2fv  26420
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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454
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