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Theorem cnvresid 5290
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 5033 . . 3 I = I
21eqcomi 2181 . 2 I = I
3 funi 5248 . . 3 Fun I
4 funeq 5236 . . 3 ( I = I → (Fun I ↔ Fun I ))
53, 4mpbii 148 . 2 ( I = I → Fun I )
6 funcnvres 5289 . . 3 (Fun I → ( I ↾ 𝐴) = ( I ↾ ( I “ 𝐴)))
7 imai 4984 . . . 4 ( I “ 𝐴) = 𝐴
81, 7reseq12i 4905 . . 3 ( I ↾ ( I “ 𝐴)) = ( I ↾ 𝐴)
96, 8eqtrdi 2226 . 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 1353   I cid 4288  ccnv 4625  cres 4628  cima 4629  Fun wfun 5210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004  df-opab 4065  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-fun 5218
This theorem is referenced by:  fcoi1  5396  f1oi  5499  ssidcn  13603  idhmeo  13710
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