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Theorem mapdom1 7231
Description: Order-preserving property of set exponentiation. Theorem 6L(c) of [Enderton] p. 149. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 9-Mar-2013.)
Assertion
Ref Expression
mapdom1  |-  ( A  ~<_  B  ->  ( A  ^m  C )  ~<_  ( B  ^m  C ) )

Proof of Theorem mapdom1
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 reldom 7074 . . . . . . 7  |-  Rel  ~<_
21brrelex2i 4878 . . . . . 6  |-  ( A  ~<_  B  ->  B  e.  _V )
3 domeng 7081 . . . . . 6  |-  ( B  e.  _V  ->  ( A  ~<_  B  <->  E. x
( A  ~~  x  /\  x  C_  B ) ) )
42, 3syl 16 . . . . 5  |-  ( A  ~<_  B  ->  ( A  ~<_  B 
<->  E. x ( A 
~~  x  /\  x  C_  B ) ) )
54ibi 233 . . . 4  |-  ( A  ~<_  B  ->  E. x
( A  ~~  x  /\  x  C_  B ) )
65adantr 452 . . 3  |-  ( ( A  ~<_  B  /\  C  e.  _V )  ->  E. x
( A  ~~  x  /\  x  C_  B ) )
7 simpl 444 . . . . 5  |-  ( ( A  ~~  x  /\  x  C_  B )  ->  A  ~~  x )
8 enrefg 7098 . . . . . 6  |-  ( C  e.  _V  ->  C  ~~  C )
98adantl 453 . . . . 5  |-  ( ( A  ~<_  B  /\  C  e.  _V )  ->  C  ~~  C )
10 mapen 7230 . . . . 5  |-  ( ( A  ~~  x  /\  C  ~~  C )  -> 
( A  ^m  C
)  ~~  ( x  ^m  C ) )
117, 9, 10syl2anr 465 . . . 4  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( A  ~~  x  /\  x  C_  B ) )  ->  ( A  ^m  C )  ~~  (
x  ^m  C )
)
12 ovex 6065 . . . . 5  |-  ( B  ^m  C )  e. 
_V
132ad2antrr 707 . . . . . 6  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( A  ~~  x  /\  x  C_  B ) )  ->  B  e.  _V )
14 simprr 734 . . . . . 6  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( A  ~~  x  /\  x  C_  B ) )  ->  x  C_  B
)
15 mapss 7015 . . . . . 6  |-  ( ( B  e.  _V  /\  x  C_  B )  -> 
( x  ^m  C
)  C_  ( B  ^m  C ) )
1613, 14, 15syl2anc 643 . . . . 5  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( A  ~~  x  /\  x  C_  B ) )  ->  ( x  ^m  C )  C_  ( B  ^m  C ) )
17 ssdomg 7112 . . . . 5  |-  ( ( B  ^m  C )  e.  _V  ->  (
( x  ^m  C
)  C_  ( B  ^m  C )  ->  (
x  ^m  C )  ~<_  ( B  ^m  C ) ) )
1812, 16, 17mpsyl 61 . . . 4  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( A  ~~  x  /\  x  C_  B ) )  ->  ( x  ^m  C )  ~<_  ( B  ^m  C ) )
19 endomtr 7124 . . . 4  |-  ( ( ( A  ^m  C
)  ~~  ( x  ^m  C )  /\  (
x  ^m  C )  ~<_  ( B  ^m  C ) )  ->  ( A  ^m  C )  ~<_  ( B  ^m  C ) )
2011, 18, 19syl2anc 643 . . 3  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( A  ~~  x  /\  x  C_  B ) )  ->  ( A  ^m  C )  ~<_  ( B  ^m  C ) )
216, 20exlimddv 1645 . 2  |-  ( ( A  ~<_  B  /\  C  e.  _V )  ->  ( A  ^m  C )  ~<_  ( B  ^m  C ) )
22 elmapex 6996 . . . . . . 7  |-  ( x  e.  ( A  ^m  C )  ->  ( A  e.  _V  /\  C  e.  _V ) )
2322simprd 450 . . . . . 6  |-  ( x  e.  ( A  ^m  C )  ->  C  e.  _V )
2423con3i 129 . . . . 5  |-  ( -.  C  e.  _V  ->  -.  x  e.  ( A  ^m  C ) )
2524eq0rdv 3622 . . . 4  |-  ( -.  C  e.  _V  ->  ( A  ^m  C )  =  (/) )
2625adantl 453 . . 3  |-  ( ( A  ~<_  B  /\  -.  C  e.  _V )  ->  ( A  ^m  C
)  =  (/) )
27120dom 7196 . . 3  |-  (/)  ~<_  ( B  ^m  C )
2826, 27syl6eqbr 4209 . 2  |-  ( ( A  ~<_  B  /\  -.  C  e.  _V )  ->  ( A  ^m  C
)  ~<_  ( B  ^m  C ) )
2921, 28pm2.61dan 767 1  |-  ( A  ~<_  B  ->  ( A  ^m  C )  ~<_  ( B  ^m  C ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1721   _Vcvv 2916    C_ wss 3280   (/)c0 3588   class class class wbr 4172  (class class class)co 6040    ^m cmap 6977    ~~ cen 7065    ~<_ cdom 7066
This theorem is referenced by:  mappwen  7949  pwcfsdom  8414  cfpwsdom  8415  rpnnen  12781  rexpen  12782  hauspwdom  17517
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-map 6979  df-en 7069  df-dom 7070
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