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Theorem fvresi 5686
Description: The value of a restricted identity function. (Contributed by NM, 19-May-2004.)
Assertion
Ref Expression
fvresi  |-  ( B  e.  A  ->  (
(  _I  |`  A ) `
 B )  =  B )

Proof of Theorem fvresi
StepHypRef Expression
1 fvres 5518 . 2  |-  ( B  e.  A  ->  (
(  _I  |`  A ) `
 B )  =  (  _I  `  B
) )
2 fvi 5551 . 2  |-  ( B  e.  A  ->  (  _I  `  B )  =  B )
31, 2eqtrd 2203 1  |-  ( B  e.  A  ->  (
(  _I  |`  A ) `
 B )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1348    e. wcel 2141    _I cid 4271    |` cres 4611   ` cfv 5196
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-res 4621  df-iota 5158  df-fun 5198  df-fv 5204
This theorem is referenced by:  f1ocnvfv1  5753  f1ocnvfv2  5754  fcof1  5759  fcofo  5760  isoid  5786  iordsmo  6273  omp1eomlem  7067  ctm  7082  ndxarg  12426  idmhm  12678  dvid  13377
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