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Theorem mapdom3 7271
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 3629 . . 3  |-  ( B  =/=  (/)  <->  E. x  x  e.  B )
2 oveq1 6080 . . . . . . . . . 10  |-  ( y  =  A  ->  (
y  ^m  { x } )  =  ( A  ^m  { x } ) )
3 id 20 . . . . . . . . . 10  |-  ( y  =  A  ->  y  =  A )
42, 3breq12d 4217 . . . . . . . . 9  |-  ( y  =  A  ->  (
( y  ^m  {
x } )  ~~  y 
<->  ( A  ^m  {
x } )  ~~  A ) )
5 vex 2951 . . . . . . . . . 10  |-  y  e. 
_V
6 vex 2951 . . . . . . . . . 10  |-  x  e. 
_V
75, 6mapsnen 7176 . . . . . . . . 9  |-  ( y  ^m  { x }
)  ~~  y
84, 7vtoclg 3003 . . . . . . . 8  |-  ( A  e.  V  ->  ( A  ^m  { x }
)  ~~  A )
983ad2ant1 978 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  ( A  ^m  {
x } )  ~~  A )
109ensymd 7150 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  A  ~~  ( A  ^m  { x }
) )
11 simp2 958 . . . . . . . 8  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  B  e.  W )
12 simp3 959 . . . . . . . . 9  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  x  e.  B )
1312snssd 3935 . . . . . . . 8  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  { x }  C_  B )
14 ssdomg 7145 . . . . . . . 8  |-  ( B  e.  W  ->  ( { x }  C_  B  ->  { x }  ~<_  B ) )
1511, 13, 14sylc 58 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  { x }  ~<_  B )
166snnz 3914 . . . . . . . 8  |-  { x }  =/=  (/)
17 simpl 444 . . . . . . . . 9  |-  ( ( { x }  =  (/) 
/\  A  =  (/) )  ->  { x }  =  (/) )
1817necon3ai 2638 . . . . . . . 8  |-  ( { x }  =/=  (/)  ->  -.  ( { x }  =  (/) 
/\  A  =  (/) ) )
1916, 18ax-mp 8 . . . . . . 7  |-  -.  ( { x }  =  (/) 
/\  A  =  (/) )
20 mapdom2 7270 . . . . . . 7  |-  ( ( { x }  ~<_  B  /\  -.  ( { x }  =  (/)  /\  A  =  (/) ) )  ->  ( A  ^m  { x }
)  ~<_  ( A  ^m  B ) )
2115, 19, 20sylancl 644 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  ( A  ^m  {
x } )  ~<_  ( A  ^m  B ) )
22 endomtr 7157 . . . . . 6  |-  ( ( A  ~~  ( A  ^m  { x }
)  /\  ( A  ^m  { x } )  ~<_  ( A  ^m  B
) )  ->  A  ~<_  ( A  ^m  B ) )
2310, 21, 22syl2anc 643 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  x  e.  B )  ->  A  ~<_  ( A  ^m  B ) )
24233expia 1155 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( x  e.  B  ->  A  ~<_  ( A  ^m  B ) ) )
2524exlimdv 1646 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( E. x  x  e.  B  ->  A  ~<_  ( A  ^m  B ) ) )
261, 25syl5bi 209 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( B  =/=  (/)  ->  A  ~<_  ( A  ^m  B ) ) )
27263impia 1150 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 359    /\ w3a 936   E.wex 1550    = wceq 1652    e. wcel 1725    =/= wne 2598    C_ wss 3312   (/)c0 3620   {csn 3806   class class class wbr 4204  (class class class)co 6073    ^m cmap 7010    ~~ cen 7098    ~<_ cdom 7099
This theorem is referenced by:  infmap2  8090
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  ax-un 4693
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-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-er 6897  df-map 7012  df-en 7102  df-dom 7103
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