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Theorem fviss 6400
Description: The value of the identity function is a subset of the argument. (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 6364 . . . 4 (𝑥 ∈ ( I ‘𝐴) → 𝐴 ∈ V)
3 fvi 6399 . . . 4 (𝐴 ∈ V → ( I ‘𝐴) = 𝐴)
42, 3syl 17 . . 3 (𝑥 ∈ ( I ‘𝐴) → ( I ‘𝐴) = 𝐴)
51, 4eleqtrd 2852 . 2 (𝑥 ∈ ( I ‘𝐴) → 𝑥𝐴)
65ssriv 3756 1 ( I ‘𝐴) ⊆ 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1631  wcel 2145  Vcvv 3351  wss 3723   I cid 5157  cfv 6030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-iota 5993  df-fun 6032  df-fv 6038
This theorem is referenced by:  efglem  18336  efgtf  18342  efgtlen  18346  efginvrel2  18347  efginvrel1  18348  efgsfo  18359  efgredlemg  18362  efgredleme  18363  efgredlemd  18364  efgredlemc  18365  efgredlem  18367  efgred  18368  efgcpbllemb  18375  frgpinv  18384  frgpuplem  18392  frgpupf  18393  frgpup1  18395
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