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Theorem f1ocnvfv1 5570
Description: The converse value of the value of a one-to-one onto function. (Contributed by NM, 20-May-2004.)
Assertion
Ref Expression
f1ocnvfv1  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  A )  ->  ( `' F `  ( F `  C ) )  =  C )

Proof of Theorem f1ocnvfv1
StepHypRef Expression
1 f1ococnv1 5295 . . . 4  |-  ( F : A -1-1-onto-> B  ->  ( `' F  o.  F )  =  (  _I  |`  A ) )
21fveq1d 5320 . . 3  |-  ( F : A -1-1-onto-> B  ->  ( ( `' F  o.  F
) `  C )  =  ( (  _I  |`  A ) `  C
) )
32adantr 271 . 2  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  A )  ->  ( ( `' F  o.  F ) `  C
)  =  ( (  _I  |`  A ) `  C ) )
4 f1of 5266 . . 3  |-  ( F : A -1-1-onto-> B  ->  F : A
--> B )
5 fvco3 5388 . . 3  |-  ( ( F : A --> B  /\  C  e.  A )  ->  ( ( `' F  o.  F ) `  C
)  =  ( `' F `  ( F `
 C ) ) )
64, 5sylan 278 . 2  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  A )  ->  ( ( `' F  o.  F ) `  C
)  =  ( `' F `  ( F `
 C ) ) )
7 fvresi 5504 . . 3  |-  ( C  e.  A  ->  (
(  _I  |`  A ) `
 C )  =  C )
87adantl 272 . 2  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  A )  ->  ( (  _I  |`  A ) `
 C )  =  C )
93, 6, 83eqtr3d 2129 1  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  A )  ->  ( `' F `  ( F `  C ) )  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1290    e. wcel 1439    _I cid 4124   `'ccnv 4450    |` cres 4453    o. ccom 4455   -->wf 5024   -1-1-onto->wf1o 5027   ` cfv 5028
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-sbc 2842  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-id 4129  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-rn 4462  df-res 4463  df-ima 4464  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036
This theorem is referenced by:  f1ocnvfv  5572  iseqf1olemab  9972  cnrecnv  10398
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