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Theorem mapdom3 7033
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 3464 . . 3  |-  ( B  =/=  (/)  <->  E. x  x  e.  B )
2 oveq1 5865 . . . . . . . . . 10  |-  ( y  =  A  ->  (
y  ^m  { x } )  =  ( A  ^m  { x } ) )
3 id 19 . . . . . . . . . 10  |-  ( y  =  A  ->  y  =  A )
42, 3breq12d 4036 . . . . . . . . 9  |-  ( y  =  A  ->  (
( y  ^m  {
x } )  ~~  y 
<->  ( A  ^m  {
x } )  ~~  A ) )
5 vex 2791 . . . . . . . . . 10  |-  y  e. 
_V
6 vex 2791 . . . . . . . . . 10  |-  x  e. 
_V
75, 6mapsnen 6938 . . . . . . . . 9  |-  ( y  ^m  { x }
)  ~~  y
84, 7vtoclg 2843 . . . . . . . 8  |-  ( A  e.  V  ->  ( A  ^m  { x }
)  ~~  A )
983ad2ant1 976 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  ( A  ^m  {
x } )  ~~  A )
10 ensym 6910 . . . . . . 7  |-  ( ( A  ^m  { x } )  ~~  A  ->  A  ~~  ( A  ^m  { x }
) )
119, 10syl 15 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  A  ~~  ( A  ^m  { x }
) )
12 simp2 956 . . . . . . . 8  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  B  e.  W )
13 simp3 957 . . . . . . . . 9  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  x  e.  B )
1413snssd 3760 . . . . . . . 8  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  { x }  C_  B )
15 ssdomg 6907 . . . . . . . 8  |-  ( B  e.  W  ->  ( { x }  C_  B  ->  { x }  ~<_  B ) )
1612, 14, 15sylc 56 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  { x }  ~<_  B )
176snnz 3744 . . . . . . . 8  |-  { x }  =/=  (/)
18 simpl 443 . . . . . . . . 9  |-  ( ( { x }  =  (/) 
/\  A  =  (/) )  ->  { x }  =  (/) )
1918necon3ai 2486 . . . . . . . 8  |-  ( { x }  =/=  (/)  ->  -.  ( { x }  =  (/) 
/\  A  =  (/) ) )
2017, 19ax-mp 8 . . . . . . 7  |-  -.  ( { x }  =  (/) 
/\  A  =  (/) )
21 mapdom2 7032 . . . . . . 7  |-  ( ( { x }  ~<_  B  /\  -.  ( { x }  =  (/)  /\  A  =  (/) ) )  ->  ( A  ^m  { x }
)  ~<_  ( A  ^m  B ) )
2216, 20, 21sylancl 643 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  ( A  ^m  {
x } )  ~<_  ( A  ^m  B ) )
23 endomtr 6919 . . . . . 6  |-  ( ( A  ~~  ( A  ^m  { x }
)  /\  ( A  ^m  { x } )  ~<_  ( A  ^m  B
) )  ->  A  ~<_  ( A  ^m  B ) )
2411, 22, 23syl2anc 642 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  A  ~<_  ( A  ^m  B ) )
25243expia 1153 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( x  e.  B  ->  A  ~<_  ( A  ^m  B ) ) )
2625exlimdv 1664 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( E. x  x  e.  B  ->  A  ~<_  ( A  ^m  B ) ) )
271, 26syl5bi 208 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( B  =/=  (/)  ->  A  ~<_  ( A  ^m  B ) ) )
28273impia 1148 1  |-  ( ( A  e.  V  /\  B  e.  W  /\  B  =/=  (/) )  ->  A  ~<_  ( A  ^m  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684    =/= wne 2446    C_ wss 3152   (/)c0 3455   {csn 3640   class class class wbr 4023  (class class class)co 5858    ^m cmap 6772    ~~ cen 6860    ~<_ cdom 6861
This theorem is referenced by:  infmap2  7844
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-er 6660  df-map 6774  df-en 6864  df-dom 6865
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