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Theorem fviss 6713
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 6675 . . . 4 (𝑥 ∈ ( I ‘𝐴) → 𝐴 ∈ V)
3 fvi 6712 . . . 4 (𝐴 ∈ V → ( I ‘𝐴) = 𝐴)
42, 3syl 17 . . 3 (𝑥 ∈ ( I ‘𝐴) → ( I ‘𝐴) = 𝐴)
51, 4eleqtrd 2913 . 2 (𝑥 ∈ ( I ‘𝐴) → 𝑥𝐴)
65ssriv 3946 1 ( I ‘𝐴) ⊆ 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wcel 2114  Vcvv 3470  wss 3909   I cid 5431  cfv 6327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5175  ax-nul 5182  ax-pow 5238  ax-pr 5302
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3472  df-sbc 3749  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4811  df-br 5039  df-opab 5101  df-id 5432  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-iota 6286  df-fun 6329  df-fv 6335
This theorem is referenced by:  efglem  18817  efgtf  18823  efgtlen  18827  efginvrel2  18828  efginvrel1  18829  efgsfo  18840  efgredlemg  18843  efgredleme  18844  efgredlemd  18845  efgredlemc  18846  efgredlem  18848  efgred  18849  efgcpbllemb  18856  frgpinv  18865  frgpuplem  18873  frgpupf  18874  frgpup1  18876
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