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Theorem f1ocan1fv 26394
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 5472 . . . 4  |-  ( G : A -1-1-onto-> B  ->  G : A
--> B )
213ad2ant2 977 . . 3  |-  ( ( Fun  F  /\  G : A -1-1-onto-> B  /\  X  e.  B )  ->  G : A --> B )
3 f1ocnv 5485 . . . . . 6  |-  ( G : A -1-1-onto-> B  ->  `' G : B -1-1-onto-> A )
4 f1of 5472 . . . . . 6  |-  ( `' G : B -1-1-onto-> A  ->  `' G : B --> A )
53, 4syl 15 . . . . 5  |-  ( G : A -1-1-onto-> B  ->  `' G : B --> A )
653ad2ant2 977 . . . 4  |-  ( ( Fun  F  /\  G : A -1-1-onto-> B  /\  X  e.  B )  ->  `' G : B --> A )
7 simp3 957 . . . 4  |-  ( ( Fun  F  /\  G : A -1-1-onto-> B  /\  X  e.  B )  ->  X  e.  B )
8 ffvelrn 5663 . . . 4  |-  ( ( `' G : B --> A  /\  X  e.  B )  ->  ( `' G `  X )  e.  A
)
96, 7, 8syl2anc 642 . . 3  |-  ( ( Fun  F  /\  G : A -1-1-onto-> B  /\  X  e.  B )  ->  ( `' G `  X )  e.  A )
10 fvco3 5596 . . 3  |-  ( ( G : A --> B  /\  ( `' G `  X )  e.  A )  -> 
( ( F  o.  G ) `  ( `' G `  X ) )  =  ( F `
 ( G `  ( `' G `  X ) ) ) )
112, 9, 10syl2anc 642 . 2  |-  ( ( Fun  F  /\  G : A -1-1-onto-> B  /\  X  e.  B )  ->  (
( F  o.  G
) `  ( `' G `  X )
)  =  ( F `
 ( G `  ( `' G `  X ) ) ) )
12 f1ocnvfv2 5793 . . . 4  |-  ( ( G : A -1-1-onto-> B  /\  X  e.  B )  ->  ( G `  ( `' G `  X ) )  =  X )
13123adant1 973 . . 3  |-  ( ( Fun  F  /\  G : A -1-1-onto-> B  /\  X  e.  B )  ->  ( G `  ( `' G `  X )
)  =  X )
1413fveq2d 5529 . 2  |-  ( ( Fun  F  /\  G : A -1-1-onto-> B  /\  X  e.  B )  ->  ( F `  ( G `  ( `' G `  X ) ) )  =  ( F `  X ) )
1511, 14eqtrd 2315 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 934    = wceq 1623    e. wcel 1684   `'ccnv 4688    o. ccom 4693   Fun wfun 5249   -->wf 5251   -1-1-onto->wf1o 5254   ` cfv 5255
This theorem is referenced by:  f1ocan2fv  26395
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263
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