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Theorem mapdom3 7741
Description: Set exponentiation dominates the mantissa. (Contributed by Mario Carneiro, 30-Apr-2015.)
Assertion
Ref Expression
mapdom3  |-  ( ( A  e.  V  /\  B  e.  W  /\  B  =/=  (/) )  ->  A  ~<_  ( A  ^m  B ) )

Proof of Theorem mapdom3
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 n0 3768 . . 3  |-  ( B  =/=  (/)  <->  E. x  x  e.  B )
2 oveq1 6303 . . . . . . . . . 10  |-  ( y  =  A  ->  (
y  ^m  { x } )  =  ( A  ^m  { x } ) )
3 id 23 . . . . . . . . . 10  |-  ( y  =  A  ->  y  =  A )
42, 3breq12d 4430 . . . . . . . . 9  |-  ( y  =  A  ->  (
( y  ^m  {
x } )  ~~  y 
<->  ( A  ^m  {
x } )  ~~  A ) )
5 vex 3081 . . . . . . . . . 10  |-  y  e. 
_V
6 vex 3081 . . . . . . . . . 10  |-  x  e. 
_V
75, 6mapsnen 7645 . . . . . . . . 9  |-  ( y  ^m  { x }
)  ~~  y
84, 7vtoclg 3136 . . . . . . . 8  |-  ( A  e.  V  ->  ( A  ^m  { x }
)  ~~  A )
983ad2ant1 1026 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  ( A  ^m  {
x } )  ~~  A )
109ensymd 7618 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  A  ~~  ( A  ^m  { x }
) )
11 simp2 1006 . . . . . . . 8  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  B  e.  W )
12 simp3 1007 . . . . . . . . 9  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  x  e.  B )
1312snssd 4139 . . . . . . . 8  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  { x }  C_  B )
14 ssdomg 7613 . . . . . . . 8  |-  ( B  e.  W  ->  ( { x }  C_  B  ->  { x }  ~<_  B ) )
1511, 13, 14sylc 62 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  { x }  ~<_  B )
166snnz 4112 . . . . . . . 8  |-  { x }  =/=  (/)
17 simpl 458 . . . . . . . . 9  |-  ( ( { x }  =  (/) 
/\  A  =  (/) )  ->  { x }  =  (/) )
1817necon3ai 2650 . . . . . . . 8  |-  ( { x }  =/=  (/)  ->  -.  ( { x }  =  (/) 
/\  A  =  (/) ) )
1916, 18ax-mp 5 . . . . . . 7  |-  -.  ( { x }  =  (/) 
/\  A  =  (/) )
20 mapdom2 7740 . . . . . . 7  |-  ( ( { x }  ~<_  B  /\  -.  ( { x }  =  (/)  /\  A  =  (/) ) )  ->  ( A  ^m  { x }
)  ~<_  ( A  ^m  B ) )
2115, 19, 20sylancl 666 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  ( A  ^m  {
x } )  ~<_  ( A  ^m  B ) )
22 endomtr 7625 . . . . . 6  |-  ( ( A  ~~  ( A  ^m  { x }
)  /\  ( A  ^m  { x } )  ~<_  ( A  ^m  B
) )  ->  A  ~<_  ( A  ^m  B ) )
2310, 21, 22syl2anc 665 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  A  ~<_  ( A  ^m  B ) )
24233expia 1207 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( x  e.  B  ->  A  ~<_  ( A  ^m  B ) ) )
2524exlimdv 1768 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( E. x  x  e.  B  ->  A  ~<_  ( A  ^m  B ) ) )
261, 25syl5bi 220 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( B  =/=  (/)  ->  A  ~<_  ( A  ^m  B ) ) )
27263impia 1202 1  |-  ( ( A  e.  V  /\  B  e.  W  /\  B  =/=  (/) )  ->  A  ~<_  ( A  ^m  B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437   E.wex 1659    e. wcel 1867    =/= wne 2616    C_ wss 3433   (/)c0 3758   {csn 3993   class class class wbr 4417  (class class class)co 6296    ^m cmap 7471    ~~ cen 7565    ~<_ cdom 7566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6798  df-2nd 6799  df-er 7362  df-map 7473  df-en 7569  df-dom 7570
This theorem is referenced by:  infmap2  8637
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