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Theorem fviss 6986
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 6945 . . . 4 (𝑥 ∈ ( I ‘𝐴) → 𝐴 ∈ V)
3 fvi 6985 . . . 4 (𝐴 ∈ V → ( I ‘𝐴) = 𝐴)
42, 3syl 17 . . 3 (𝑥 ∈ ( I ‘𝐴) → ( I ‘𝐴) = 𝐴)
51, 4eleqtrd 2841 . 2 (𝑥 ∈ ( I ‘𝐴) → 𝑥𝐴)
65ssriv 3999 1 ( I ‘𝐴) ⊆ 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wcel 2106  Vcvv 3478  wss 3963   I cid 5582  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571
This theorem is referenced by:  efglem  19749  efgtf  19755  efgtlen  19759  efginvrel2  19760  efginvrel1  19761  efgsfo  19772  efgredlemg  19775  efgredleme  19776  efgredlemd  19777  efgredlemc  19778  efgredlem  19780  efgred  19781  efgcpbllemb  19788  frgpinv  19797  frgpuplem  19805  frgpupf  19806  frgpup1  19808
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