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Theorem fviss 5923
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 22 . . 3  |-  ( x  e.  (  _I  `  A )  ->  x  e.  (  _I  `  A
) )
2 elfvex 5891 . . . 4  |-  ( x  e.  (  _I  `  A )  ->  A  e.  _V )
3 fvi 5922 . . . 4  |-  ( A  e.  _V  ->  (  _I  `  A )  =  A )
42, 3syl 16 . . 3  |-  ( x  e.  (  _I  `  A )  ->  (  _I  `  A )  =  A )
51, 4eleqtrd 2557 . 2  |-  ( x  e.  (  _I  `  A )  ->  x  e.  A )
65ssriv 3508 1  |-  (  _I 
`  A )  C_  A
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379    e. wcel 1767   _Vcvv 3113    C_ wss 3476    _I cid 4790   ` cfv 5586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fv 5594
This theorem is referenced by:  efglem  16530  efgtf  16536  efgtlen  16540  efginvrel2  16541  efginvrel1  16542  efgsfo  16553  efgredlemg  16556  efgredleme  16557  efgredlemd  16558  efgredlemc  16559  efgredlem  16561  efgred  16562  efgcpbllemb  16569  frgpinv  16578  frgpuplem  16586  frgpupf  16587  frgpup1  16589
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