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Theorem fviss 5580
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 )  C_  A

Proof of Theorem fviss
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 id 19 . . 3  |-  ( x  e.  (  _I  `  A )  ->  x  e.  (  _I  `  A
) )
2 elfvex 5555 . . . 4  |-  ( x  e.  (  _I  `  A )  ->  A  e.  _V )
3 fvi 5579 . . . 4  |-  ( A  e.  _V  ->  (  _I  `  A )  =  A )
42, 3syl 15 . . 3  |-  ( x  e.  (  _I  `  A )  ->  (  _I  `  A )  =  A )
51, 4eleqtrd 2359 . 2  |-  ( x  e.  (  _I  `  A )  ->  x  e.  A )
65ssriv 3184 1  |-  (  _I 
`  A )  C_  A
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152    _I cid 4304   ` cfv 5255
This theorem is referenced by:  efglem  15025  efgtf  15031  efgtlen  15035  efginvrel2  15036  efginvrel1  15037  efgsfo  15048  efgredlemg  15051  efgredleme  15052  efgredlemd  15053  efgredlemc  15054  efgredlem  15056  efgred  15057  efgcpbllemb  15064  frgpinv  15073  frgpuplem  15081  frgpupf  15082  frgpup1  15084
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263
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