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Theorem cnvresid 4973
 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 4728 . . 3 I = I
21eqcomi 2044 . 2 I = I
3 funi 4932 . . 3 Fun I
4 funeq 4921 . . 3 ( I = I → (Fun I ↔ Fun I ))
53, 4mpbii 136 . 2 ( I = I → Fun I )
6 funcnvres 4972 . . 3 (Fun I → ( I ↾ 𝐴) = ( I ↾ ( I “ 𝐴)))
7 imai 4681 . . . 4 ( I “ 𝐴) = 𝐴
81, 7reseq12i 4610 . . 3 ( I ↾ ( I “ 𝐴)) = ( I ↾ 𝐴)
96, 8syl6eq 2088 . 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 1243   I cid 4025  ◡ccnv 4344   ↾ cres 4347   “ cima 4348  Fun wfun 4896 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-fun 4904 This theorem is referenced by:  fcoi1  5070  f1oi  5164
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