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Theorem mapdom3 7690
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 3794 . . 3  |-  ( B  =/=  (/)  <->  E. x  x  e.  B )
2 oveq1 6292 . . . . . . . . . 10  |-  ( y  =  A  ->  (
y  ^m  { x } )  =  ( A  ^m  { x } ) )
3 id 22 . . . . . . . . . 10  |-  ( y  =  A  ->  y  =  A )
42, 3breq12d 4460 . . . . . . . . 9  |-  ( y  =  A  ->  (
( y  ^m  {
x } )  ~~  y 
<->  ( A  ^m  {
x } )  ~~  A ) )
5 vex 3116 . . . . . . . . . 10  |-  y  e. 
_V
6 vex 3116 . . . . . . . . . 10  |-  x  e. 
_V
75, 6mapsnen 7594 . . . . . . . . 9  |-  ( y  ^m  { x }
)  ~~  y
84, 7vtoclg 3171 . . . . . . . 8  |-  ( A  e.  V  ->  ( A  ^m  { x }
)  ~~  A )
983ad2ant1 1017 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  ( A  ^m  {
x } )  ~~  A )
109ensymd 7567 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  A  ~~  ( A  ^m  { x }
) )
11 simp2 997 . . . . . . . 8  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  B  e.  W )
12 simp3 998 . . . . . . . . 9  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  x  e.  B )
1312snssd 4172 . . . . . . . 8  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  { x }  C_  B )
14 ssdomg 7562 . . . . . . . 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 4145 . . . . . . . 8  |-  { x }  =/=  (/)
17 simpl 457 . . . . . . . . 9  |-  ( ( { x }  =  (/) 
/\  A  =  (/) )  ->  { x }  =  (/) )
1817necon3ai 2695 . . . . . . . 8  |-  ( { x }  =/=  (/)  ->  -.  ( { x }  =  (/) 
/\  A  =  (/) ) )
1916, 18ax-mp 5 . . . . . . 7  |-  -.  ( { x }  =  (/) 
/\  A  =  (/) )
20 mapdom2 7689 . . . . . . 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 7574 . . . . . 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 1198 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( x  e.  B  ->  A  ~<_  ( A  ^m  B ) ) )
2524exlimdv 1700 . . 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 1193 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 973    = wceq 1379   E.wex 1596    e. wcel 1767    =/= wne 2662    C_ wss 3476   (/)c0 3785   {csn 4027   class class class wbr 4447  (class class class)co 6285    ^m cmap 7421    ~~ cen 7514    ~<_ cdom 7515
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-1st 6785  df-2nd 6786  df-er 7312  df-map 7423  df-en 7518  df-dom 7519
This theorem is referenced by:  infmap2  8599
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