MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fviss Structured version   Visualization version   GIF version

Theorem fviss 6213
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 6178 . . . 4 (𝑥 ∈ ( I ‘𝐴) → 𝐴 ∈ V)
3 fvi 6212 . . . 4 (𝐴 ∈ V → ( I ‘𝐴) = 𝐴)
42, 3syl 17 . . 3 (𝑥 ∈ ( I ‘𝐴) → ( I ‘𝐴) = 𝐴)
51, 4eleqtrd 2700 . 2 (𝑥 ∈ ( I ‘𝐴) → 𝑥𝐴)
65ssriv 3587 1 ( I ‘𝐴) ⊆ 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  wcel 1987  Vcvv 3186  wss 3555   I cid 4984  cfv 5847
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855
This theorem is referenced by:  efglem  18050  efgtf  18056  efgtlen  18060  efginvrel2  18061  efginvrel1  18062  efgsfo  18073  efgredlemg  18076  efgredleme  18077  efgredlemd  18078  efgredlemc  18079  efgredlem  18081  efgred  18082  efgcpbllemb  18089  frgpinv  18098  frgpuplem  18106  frgpupf  18107  frgpup1  18109
  Copyright terms: Public domain W3C validator