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Theorem fviss 5776
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 20 . . 3  |-  ( x  e.  (  _I  `  A )  ->  x  e.  (  _I  `  A
) )
2 elfvex 5750 . . . 4  |-  ( x  e.  (  _I  `  A )  ->  A  e.  _V )
3 fvi 5775 . . . 4  |-  ( A  e.  _V  ->  (  _I  `  A )  =  A )
42, 3syl 16 . . 3  |-  ( x  e.  (  _I  `  A )  ->  (  _I  `  A )  =  A )
51, 4eleqtrd 2511 . 2  |-  ( x  e.  (  _I  `  A )  ->  x  e.  A )
65ssriv 3344 1  |-  (  _I 
`  A )  C_  A
Colors of variables: wff set class
Syntax hints:    = wceq 1652    e. wcel 1725   _Vcvv 2948    C_ wss 3312    _I cid 4485   ` cfv 5446
This theorem is referenced by:  efglem  15340  efgtf  15346  efgtlen  15350  efginvrel2  15351  efginvrel1  15352  efgsfo  15363  efgredlemg  15366  efgredleme  15367  efgredlemd  15368  efgredlemc  15369  efgredlem  15371  efgred  15372  efgcpbllemb  15379  frgpinv  15388  frgpuplem  15396  frgpupf  15397  frgpup1  15399
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454
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