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Theorem mapdom3 7594
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 3755 . . 3  |-  ( B  =/=  (/)  <->  E. x  x  e.  B )
2 oveq1 6208 . . . . . . . . . 10  |-  ( y  =  A  ->  (
y  ^m  { x } )  =  ( A  ^m  { x } ) )
3 id 22 . . . . . . . . . 10  |-  ( y  =  A  ->  y  =  A )
42, 3breq12d 4414 . . . . . . . . 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 7498 . . . . . . . . 9  |-  ( y  ^m  { x }
)  ~~  y
84, 7vtoclg 3136 . . . . . . . 8  |-  ( A  e.  V  ->  ( A  ^m  { x }
)  ~~  A )
983ad2ant1 1009 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  ( A  ^m  {
x } )  ~~  A )
109ensymd 7471 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  A  ~~  ( A  ^m  { x }
) )
11 simp2 989 . . . . . . . 8  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  B  e.  W )
12 simp3 990 . . . . . . . . 9  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  x  e.  B )
1312snssd 4127 . . . . . . . 8  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  { x }  C_  B )
14 ssdomg 7466 . . . . . . . 8  |-  ( B  e.  W  ->  ( { x }  C_  B  ->  { x }  ~<_  B ) )
1511, 13, 14sylc 60 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  { x }  ~<_  B )
166snnz 4102 . . . . . . . 8  |-  { x }  =/=  (/)
17 simpl 457 . . . . . . . . 9  |-  ( ( { x }  =  (/) 
/\  A  =  (/) )  ->  { x }  =  (/) )
1817necon3ai 2680 . . . . . . . 8  |-  ( { x }  =/=  (/)  ->  -.  ( { x }  =  (/) 
/\  A  =  (/) ) )
1916, 18ax-mp 5 . . . . . . 7  |-  -.  ( { x }  =  (/) 
/\  A  =  (/) )
20 mapdom2 7593 . . . . . . 7  |-  ( ( { x }  ~<_  B  /\  -.  ( { x }  =  (/)  /\  A  =  (/) ) )  ->  ( A  ^m  { x }
)  ~<_  ( A  ^m  B ) )
2115, 19, 20sylancl 662 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  ( A  ^m  {
x } )  ~<_  ( A  ^m  B ) )
22 endomtr 7478 . . . . . 6  |-  ( ( A  ~~  ( A  ^m  { x }
)  /\  ( A  ^m  { x } )  ~<_  ( A  ^m  B
) )  ->  A  ~<_  ( A  ^m  B ) )
2310, 21, 22syl2anc 661 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  A  ~<_  ( A  ^m  B ) )
24233expia 1190 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( x  e.  B  ->  A  ~<_  ( A  ^m  B ) ) )
2524exlimdv 1691 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( E. x  x  e.  B  ->  A  ~<_  ( A  ^m  B ) ) )
261, 25syl5bi 217 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( B  =/=  (/)  ->  A  ~<_  ( A  ^m  B ) ) )
27263impia 1185 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 369    /\ w3a 965    = wceq 1370   E.wex 1587    e. wcel 1758    =/= wne 2648    C_ wss 3437   (/)c0 3746   {csn 3986   class class class wbr 4401  (class class class)co 6201    ^m cmap 7325    ~~ cen 7418    ~<_ cdom 7419
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-1st 6688  df-2nd 6689  df-er 7212  df-map 7327  df-en 7422  df-dom 7423
This theorem is referenced by:  infmap2  8499
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