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Theorem fviss 5596
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 5571 . . . 4  |-  ( x  e.  (  _I  `  A )  ->  A  e.  _V )
3 fvi 5595 . . . 4  |-  ( A  e.  _V  ->  (  _I  `  A )  =  A )
42, 3syl 15 . . 3  |-  ( x  e.  (  _I  `  A )  ->  (  _I  `  A )  =  A )
51, 4eleqtrd 2372 . 2  |-  ( x  e.  (  _I  `  A )  ->  x  e.  A )
65ssriv 3197 1  |-  (  _I 
`  A )  C_  A
Colors of variables: wff set class
Syntax hints:    = wceq 1632    e. wcel 1696   _Vcvv 2801    C_ wss 3165    _I cid 4320   ` cfv 5271
This theorem is referenced by:  efglem  15041  efgtf  15047  efgtlen  15051  efginvrel2  15052  efginvrel1  15053  efgsfo  15064  efgredlemg  15067  efgredleme  15068  efgredlemd  15069  efgredlemc  15070  efgredlem  15072  efgred  15073  efgcpbllemb  15080  frgpinv  15089  frgpuplem  15097  frgpupf  15098  frgpup1  15100
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279
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