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Theorem cnvresid 5001
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 4756 . . 3 I = I
21eqcomi 2060 . 2 I = I
3 funi 4960 . . 3 Fun I
4 funeq 4949 . . 3 ( I = I → (Fun I ↔ Fun I ))
53, 4mpbii 140 . 2 ( I = I → Fun I )
6 funcnvres 5000 . . 3 (Fun I → ( I ↾ 𝐴) = ( I ↾ ( I “ 𝐴)))
7 imai 4709 . . . 4 ( I “ 𝐴) = 𝐴
81, 7reseq12i 4638 . . 3 ( I ↾ ( I “ 𝐴)) = ( I ↾ 𝐴)
96, 8syl6eq 2104 . 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 1259   I cid 4053  ccnv 4372  cres 4375  cima 4376  Fun wfun 4924
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-fun 4932
This theorem is referenced by:  fcoi1  5098  f1oi  5192
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