ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  cnvresid Structured version   GIF version

Theorem cnvresid 4916
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 4671 . . 3 I = I
21eqcomi 2041 . 2 I = I
3 funi 4875 . . 3 Fun I
4 funeq 4864 . . 3 ( I = I → (Fun I ↔ Fun I ))
53, 4mpbii 136 . 2 ( I = I → Fun I )
6 funcnvres 4915 . . 3 (Fun I → ( I ↾ A) = ( I ↾ ( I “ A)))
7 imai 4624 . . . 4 ( I “ A) = A
81, 7reseq12i 4553 . . 3 ( I ↾ ( I “ A)) = ( I ↾ A)
96, 8syl6eq 2085 . 2 (Fun I → ( I ↾ A) = ( I ↾ A))
102, 5, 9mp2b 8 1 ( I ↾ A) = ( I ↾ A)
Colors of variables: wff set class
Syntax hints:   = wceq 1242   I cid 4016  ccnv 4287  cres 4290  cima 4291  Fun wfun 4839
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-fun 4847
This theorem is referenced by:  fcoi1  5013  f1oi  5107
  Copyright terms: Public domain W3C validator