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Theorem cnvresid 6645
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 6161 . . 3 I = I
21eqcomi 2746 . 2 I = I
3 funi 6598 . . 3 Fun I
4 funeq 6586 . . 3 ( I = I → (Fun I ↔ Fun I ))
53, 4mpbii 233 . 2 ( I = I → Fun I )
6 funcnvres 6644 . . 3 (Fun I → ( I ↾ 𝐴) = ( I ↾ ( I “ 𝐴)))
7 imai 6092 . . . 4 ( I “ 𝐴) = 𝐴
81, 7reseq12i 5995 . . 3 ( I ↾ ( I “ 𝐴)) = ( I ↾ 𝐴)
96, 8eqtrdi 2793 . 2 (Fun I → ( I ↾ 𝐴) = ( I ↾ 𝐴))
102, 5, 9mp2b 10 1 ( I ↾ 𝐴) = ( I ↾ 𝐴)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540   I cid 5577  ccnv 5684  cres 5687  cima 5688  Fun wfun 6555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-fun 6563
This theorem is referenced by:  fcoi1  6782  f1oi  6886  relexpcnv  15074  tsrdir  18649  gicref  19290  ssidcn  23263  idqtop  23714  idhmeo  23781  bj-iminvid  37196  ltrncnvnid  40129  dihmeetlem1N  41292  dihglblem5apreN  41293  diophrw  42770  cnvrcl0  43638  relexpaddss  43731
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