MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cnvresid Unicode version

Theorem cnvresid 5463
Description: Converse of a restricted identity function. (Contributed by FL, 4-Mar-2007.)
Assertion
Ref Expression
cnvresid  |-  `' (  _I  |`  A )  =  (  _I  |`  A )

Proof of Theorem cnvresid
StepHypRef Expression
1 cnvi 5216 . . 3  |-  `'  _I  =  _I
21eqcomi 2391 . 2  |-  _I  =  `'  _I
3 funi 5423 . . 3  |-  Fun  _I
4 funeq 5413 . . 3  |-  (  _I  =  `'  _I  ->  ( Fun  _I  <->  Fun  `'  _I  ) )
53, 4mpbii 203 . 2  |-  (  _I  =  `'  _I  ->  Fun  `'  _I  )
6 funcnvres 5462 . . 3  |-  ( Fun  `'  _I  ->  `' (  _I  |`  A )  =  ( `'  _I  |`  (  _I  " A ) ) )
7 imai 5158 . . . 4  |-  (  _I  " A )  =  A
81, 7reseq12i 5084 . . 3  |-  ( `'  _I  |`  (  _I  " A ) )  =  (  _I  |`  A )
96, 8syl6eq 2435 . 2  |-  ( Fun  `'  _I  ->  `' (  _I  |`  A )  =  (  _I  |`  A ) )
102, 5, 9mp2b 10 1  |-  `' (  _I  |`  A )  =  (  _I  |`  A )
Colors of variables: wff set class
Syntax hints:    = wceq 1649    _I cid 4434   `'ccnv 4817    |` cres 4820   "cima 4821   Fun wfun 5388
This theorem is referenced by:  fcoi1  5557  f1oi  5653  tsrdir  14610  gicref  14985  ssidcn  17241  idqtop  17659  idhmeo  17726  relexpcnv  24912  diophrw  26508  ltrncnvnid  30241  dihmeetlem1N  31405  dihglblem5apreN  31406
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-br 4154  df-opab 4208  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-fun 5396
  Copyright terms: Public domain W3C validator