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Theorem cnvresid 5088
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 4836 . . 3 I = I
21eqcomi 2092 . 2 I = I
3 funi 5046 . . 3 Fun I
4 funeq 5035 . . 3 ( I = I → (Fun I ↔ Fun I ))
53, 4mpbii 146 . 2 ( I = I → Fun I )
6 funcnvres 5087 . . 3 (Fun I → ( I ↾ 𝐴) = ( I ↾ ( I “ 𝐴)))
7 imai 4788 . . . 4 ( I “ 𝐴) = 𝐴
81, 7reseq12i 4711 . . 3 ( I ↾ ( I “ 𝐴)) = ( I ↾ 𝐴)
96, 8syl6eq 2136 . 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 1289   I cid 4115  ccnv 4437  cres 4440  cima 4441  Fun wfun 5009
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-fun 5017
This theorem is referenced by:  fcoi1  5191  f1oi  5291
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