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Theorem f1ocnvfv1 5778
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 5491 . . . 4  |-  ( F : A -1-1-onto-> B  ->  ( `' F  o.  F )  =  (  _I  |`  A ) )
21fveq1d 5518 . . 3  |-  ( F : A -1-1-onto-> B  ->  ( ( `' F  o.  F
) `  C )  =  ( (  _I  |`  A ) `  C
) )
32adantr 276 . 2  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  A )  ->  ( ( `' F  o.  F ) `  C
)  =  ( (  _I  |`  A ) `  C ) )
4 f1of 5462 . . 3  |-  ( F : A -1-1-onto-> B  ->  F : A
--> B )
5 fvco3 5588 . . 3  |-  ( ( F : A --> B  /\  C  e.  A )  ->  ( ( `' F  o.  F ) `  C
)  =  ( `' F `  ( F `
 C ) ) )
64, 5sylan 283 . 2  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  A )  ->  ( ( `' F  o.  F ) `  C
)  =  ( `' F `  ( F `
 C ) ) )
7 fvresi 5710 . . 3  |-  ( C  e.  A  ->  (
(  _I  |`  A ) `
 C )  =  C )
87adantl 277 . 2  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  A )  ->  ( (  _I  |`  A ) `
 C )  =  C )
93, 6, 83eqtr3d 2218 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 104    = wceq 1353    e. wcel 2148    _I cid 4289   `'ccnv 4626    |` cres 4629    o. ccom 4631   -->wf 5213   -1-1-onto->wf1o 5216   ` cfv 5217
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-sbc 2964  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225
This theorem is referenced by:  f1ocnvfv  5780  caseinl  7090  caseinr  7091  ctssdccl  7110  cc3  7267  iseqf1olemab  10489  cnrecnv  10919  fprodssdc  11598  ennnfonelemhf1o  12414  ennnfonelemex  12415  ennnfonelemrn  12420  ctinfomlemom  12428  ssnnctlemct  12447  mhmf1o  12861  isomninnlem  14781  iswomninnlem  14800  ismkvnnlem  14803
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