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Theorem cnvresid 6429
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 5997 . . 3 I = I
21eqcomi 2834 . 2 I = I
3 funi 6383 . . 3 Fun I
4 funeq 6371 . . 3 ( I = I → (Fun I ↔ Fun I ))
53, 4mpbii 234 . 2 ( I = I → Fun I )
6 funcnvres 6428 . . 3 (Fun I → ( I ↾ 𝐴) = ( I ↾ ( I “ 𝐴)))
7 imai 5939 . . . 4 ( I “ 𝐴) = 𝐴
81, 7reseq12i 5849 . . 3 ( I ↾ ( I “ 𝐴)) = ( I ↾ 𝐴)
96, 8syl6eq 2876 . 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 1530   I cid 5457  ccnv 5552  cres 5555  cima 5556  Fun wfun 6345
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-br 5063  df-opab 5125  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-fun 6353
This theorem is referenced by:  fcoi1  6548  f1oi  6648  relexpcnv  14387  tsrdir  17840  gicref  18343  ssidcn  21781  idqtop  22232  idhmeo  22299  ltrncnvnid  37132  dihmeetlem1N  38295  dihglblem5apreN  38296  diophrw  39223  cnvrcl0  39852  relexpaddss  39930
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