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Theorem fvi 5442
 Description: The value of the identity function. (Contributed by set.mm contributors, 1-May-2004.)
Assertion
Ref Expression
fvi (A V → ( I ‘A) = A)

Proof of Theorem fvi
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 fveq2 5328 . . 3 (x = A → ( I ‘x) = ( I ‘A))
2 id 19 . . 3 (x = Ax = A)
31, 2eqeq12d 2367 . 2 (x = A → (( I ‘x) = x ↔ ( I ‘A) = A))
4 funi 5137 . . . 4 Fun I
5 dmi 4919 . . . 4 dom I = V
6 df-fn 4790 . . . 4 ( I Fn V ↔ (Fun I dom I = V))
74, 5, 6mpbir2an 886 . . 3 I Fn V
8 vex 2862 . . 3 x V
9 equid 1676 . . . . 5 x = x
108ideq 4870 . . . . . 6 (x I xx = x)
11 df-br 4640 . . . . . 6 (x I xx, x I )
1210, 11bitr3i 242 . . . . 5 (x = xx, x I )
139, 12mpbi 199 . . . 4 x, x I
14 fnopfvb 5359 . . . 4 (( I Fn V x V) → (( I ‘x) = xx, x I ))
1513, 14mpbiri 224 . . 3 (( I Fn V x V) → ( I ‘x) = x)
167, 8, 15mp2an 653 . 2 ( I ‘x) = x
173, 16vtoclg 2914 1 (A V → ( I ‘A) = A)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710  Vcvv 2859  ⟨cop 4561   class class class wbr 4639   I cid 4763  dom cdm 4772  Fun wfun 4775   Fn wfn 4776   ‘cfv 4781 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-id 4767  df-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fn 4790  df-fv 4795 This theorem is referenced by:  fvresi  5443  fvmpti  5699  fvmpt2  5704
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