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Theorem f1ocnvfv2 5449
Description: The value of the converse value of a one-to-one onto function. (Contributed by NM, 20-May-2004.)
Assertion
Ref Expression
f1ocnvfv2  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  B )  ->  ( F `  ( `' F `  C ) )  =  C )

Proof of Theorem f1ocnvfv2
StepHypRef Expression
1 f1ococnv2 5184 . . . 4  |-  ( F : A -1-1-onto-> B  ->  ( F  o.  `' F )  =  (  _I  |`  B )
)
21fveq1d 5211 . . 3  |-  ( F : A -1-1-onto-> B  ->  ( ( F  o.  `' F
) `  C )  =  ( (  _I  |`  B ) `  C
) )
32adantr 270 . 2  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  B )  ->  ( ( F  o.  `' F ) `  C
)  =  ( (  _I  |`  B ) `  C ) )
4 f1ocnv 5170 . . . 4  |-  ( F : A -1-1-onto-> B  ->  `' F : B -1-1-onto-> A )
5 f1of 5157 . . . 4  |-  ( `' F : B -1-1-onto-> A  ->  `' F : B --> A )
64, 5syl 14 . . 3  |-  ( F : A -1-1-onto-> B  ->  `' F : B --> A )
7 fvco3 5276 . . 3  |-  ( ( `' F : B --> A  /\  C  e.  B )  ->  ( ( F  o.  `' F ) `  C
)  =  ( F `
 ( `' F `  C ) ) )
86, 7sylan 277 . 2  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  B )  ->  ( ( F  o.  `' F ) `  C
)  =  ( F `
 ( `' F `  C ) ) )
9 fvresi 5388 . . 3  |-  ( C  e.  B  ->  (
(  _I  |`  B ) `
 C )  =  C )
109adantl 271 . 2  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  B )  ->  ( (  _I  |`  B ) `
 C )  =  C )
113, 8, 103eqtr3d 2122 1  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  B )  ->  ( F `  ( `' F `  C ) )  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434    _I cid 4051   `'ccnv 4370    |` cres 4373    o. ccom 4375   -->wf 4928   -1-1-onto->wf1o 4931   ` cfv 4932
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-sbc 2817  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940
This theorem is referenced by:  f1ocnvfvb  5451  isocnv  5482  f1oiso2  5497  ordiso2  6505  frecuzrdglem  9493  frecuzrdgsuc  9496  frecuzrdgdomlem  9499  frecuzrdgsuctlem  9505  frecfzennn  9508  sizefz1  9807  sqpweven  10697  2sqpwodd  10698
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