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Theorem fvi 5262
Description: The value of the identity function. (Contributed by NM, 1-May-2004.) (Revised by Mario Carneiro, 28-Apr-2015.)
Assertion
Ref Expression
fvi (𝐴𝑉 → ( I ‘𝐴) = 𝐴)

Proof of Theorem fvi
StepHypRef Expression
1 funi 4962 . 2 Fun I
2 ididg 4517 . 2 (𝐴𝑉𝐴 I 𝐴)
3 funbrfv 5244 . 2 (Fun I → (𝐴 I 𝐴 → ( I ‘𝐴) = 𝐴))
41, 2, 3mpsyl 64 1 (𝐴𝑉 → ( I ‘𝐴) = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1285  wcel 1434   class class class wbr 3793   I cid 4051  Fun wfun 4926  cfv 4932
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-sbc 2817  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-iota 4897  df-fun 4934  df-fv 4940
This theorem is referenced by:  fvresi  5388  facnn  9751  fac0  9752  fac1  9753  facp1  9754  ibcval5  9787  bcn2  9788  climshft2  10283
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