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Theorem fviss 5746
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 5714 . . . 4  |-  ( x  e.  (  _I  `  A )  ->  A  e.  _V )
3 fvi 5745 . . . 4  |-  ( A  e.  _V  ->  (  _I  `  A )  =  A )
42, 3syl 16 . . 3  |-  ( x  e.  (  _I  `  A )  ->  (  _I  `  A )  =  A )
51, 4eleqtrd 2517 . 2  |-  ( x  e.  (  _I  `  A )  ->  x  e.  A )
65ssriv 3357 1  |-  (  _I 
`  A )  C_  A
Colors of variables: wff setvar class
Syntax hints:    = wceq 1364    e. wcel 1761   _Vcvv 2970    C_ wss 3325    _I cid 4627   ` cfv 5415
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-iota 5378  df-fun 5417  df-fv 5423
This theorem is referenced by:  efglem  16206  efgtf  16212  efgtlen  16216  efginvrel2  16217  efginvrel1  16218  efgsfo  16229  efgredlemg  16232  efgredleme  16233  efgredlemd  16234  efgredlemc  16235  efgredlem  16237  efgred  16238  efgcpbllemb  16245  frgpinv  16254  frgpuplem  16262  frgpupf  16263  frgpup1  16265
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