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Theorem fviss 6985
Description: The value of the identity function is a subset of the argument. (An artifact of our function value definition.) (Contributed by Mario Carneiro, 27-Feb-2016.)
Assertion
Ref Expression
fviss ( I ‘𝐴) ⊆ 𝐴

Proof of Theorem fviss
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 id 22 . . 3 (𝑥 ∈ ( I ‘𝐴) → 𝑥 ∈ ( I ‘𝐴))
2 elfvex 6943 . . . 4 (𝑥 ∈ ( I ‘𝐴) → 𝐴 ∈ V)
3 fvi 6984 . . . 4 (𝐴 ∈ V → ( I ‘𝐴) = 𝐴)
42, 3syl 17 . . 3 (𝑥 ∈ ( I ‘𝐴) → ( I ‘𝐴) = 𝐴)
51, 4eleqtrd 2842 . 2 (𝑥 ∈ ( I ‘𝐴) → 𝑥𝐴)
65ssriv 3986 1 ( I ‘𝐴) ⊆ 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  wcel 2107  Vcvv 3479  wss 3950   I cid 5576  cfv 6560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-iota 6513  df-fun 6562  df-fv 6568
This theorem is referenced by:  efglem  19735  efgtf  19741  efgtlen  19745  efginvrel2  19746  efginvrel1  19747  efgsfo  19758  efgredlemg  19761  efgredleme  19762  efgredlemd  19763  efgredlemc  19764  efgredlem  19766  efgred  19767  efgcpbllemb  19774  frgpinv  19783  frgpuplem  19791  frgpupf  19792  frgpup1  19794
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