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Theorem mapdom1 7026
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 6869 . . . . . . 7  |-  Rel  ~<_
21brrelex2i 4730 . . . . . 6  |-  ( A  ~<_  B  ->  B  e.  _V )
3 domeng 6876 . . . . . 6  |-  ( B  e.  _V  ->  ( A  ~<_  B  <->  E. x
( A  ~~  x  /\  x  C_  B ) ) )
42, 3syl 15 . . . . 5  |-  ( A  ~<_  B  ->  ( A  ~<_  B 
<->  E. x ( A 
~~  x  /\  x  C_  B ) ) )
54ibi 232 . . . 4  |-  ( A  ~<_  B  ->  E. x
( A  ~~  x  /\  x  C_  B ) )
65adantr 451 . . 3  |-  ( ( A  ~<_  B  /\  C  e.  _V )  ->  E. x
( A  ~~  x  /\  x  C_  B ) )
7 simpl 443 . . . . . . 7  |-  ( ( A  ~~  x  /\  x  C_  B )  ->  A  ~~  x )
8 enrefg 6893 . . . . . . . 8  |-  ( C  e.  _V  ->  C  ~~  C )
98adantl 452 . . . . . . 7  |-  ( ( A  ~<_  B  /\  C  e.  _V )  ->  C  ~~  C )
10 mapen 7025 . . . . . . 7  |-  ( ( A  ~~  x  /\  C  ~~  C )  -> 
( A  ^m  C
)  ~~  ( x  ^m  C ) )
117, 9, 10syl2anr 464 . . . . . 6  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( A  ~~  x  /\  x  C_  B ) )  ->  ( A  ^m  C )  ~~  (
x  ^m  C )
)
12 ovex 5883 . . . . . . 7  |-  ( B  ^m  C )  e. 
_V
132ad2antrr 706 . . . . . . . 8  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( A  ~~  x  /\  x  C_  B ) )  ->  B  e.  _V )
14 simprr 733 . . . . . . . 8  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( A  ~~  x  /\  x  C_  B ) )  ->  x  C_  B
)
15 mapss 6810 . . . . . . . 8  |-  ( ( B  e.  _V  /\  x  C_  B )  -> 
( x  ^m  C
)  C_  ( B  ^m  C ) )
1613, 14, 15syl2anc 642 . . . . . . 7  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( A  ~~  x  /\  x  C_  B ) )  ->  ( x  ^m  C )  C_  ( B  ^m  C ) )
17 ssdomg 6907 . . . . . . 7  |-  ( ( B  ^m  C )  e.  _V  ->  (
( x  ^m  C
)  C_  ( B  ^m  C )  ->  (
x  ^m  C )  ~<_  ( B  ^m  C ) ) )
1812, 16, 17mpsyl 59 . . . . . 6  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( A  ~~  x  /\  x  C_  B ) )  ->  ( x  ^m  C )  ~<_  ( B  ^m  C ) )
19 endomtr 6919 . . . . . 6  |-  ( ( ( A  ^m  C
)  ~~  ( x  ^m  C )  /\  (
x  ^m  C )  ~<_  ( B  ^m  C ) )  ->  ( A  ^m  C )  ~<_  ( B  ^m  C ) )
2011, 18, 19syl2anc 642 . . . . 5  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( A  ~~  x  /\  x  C_  B ) )  ->  ( A  ^m  C )  ~<_  ( B  ^m  C ) )
2120ex 423 . . . 4  |-  ( ( A  ~<_  B  /\  C  e.  _V )  ->  (
( A  ~~  x  /\  x  C_  B )  ->  ( A  ^m  C )  ~<_  ( B  ^m  C ) ) )
2221exlimdv 1664 . . 3  |-  ( ( A  ~<_  B  /\  C  e.  _V )  ->  ( E. x ( A  ~~  x  /\  x  C_  B
)  ->  ( A  ^m  C )  ~<_  ( B  ^m  C ) ) )
236, 22mpd 14 . 2  |-  ( ( A  ~<_  B  /\  C  e.  _V )  ->  ( A  ^m  C )  ~<_  ( B  ^m  C ) )
24 elmapex 6791 . . . . . . 7  |-  ( x  e.  ( A  ^m  C )  ->  ( A  e.  _V  /\  C  e.  _V ) )
2524simprd 449 . . . . . 6  |-  ( x  e.  ( A  ^m  C )  ->  C  e.  _V )
2625con3i 127 . . . . 5  |-  ( -.  C  e.  _V  ->  -.  x  e.  ( A  ^m  C ) )
2726eq0rdv 3489 . . . 4  |-  ( -.  C  e.  _V  ->  ( A  ^m  C )  =  (/) )
2827adantl 452 . . 3  |-  ( ( A  ~<_  B  /\  -.  C  e.  _V )  ->  ( A  ^m  C
)  =  (/) )
29120dom 6991 . . 3  |-  (/)  ~<_  ( B  ^m  C )
3028, 29syl6eqbr 4060 . 2  |-  ( ( A  ~<_  B  /\  -.  C  e.  _V )  ->  ( A  ^m  C
)  ~<_  ( B  ^m  C ) )
3123, 30pm2.61dan 766 1  |-  ( A  ~<_  B  ->  ( A  ^m  C )  ~<_  ( B  ^m  C ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   (/)c0 3455   class class class wbr 4023  (class class class)co 5858    ^m cmap 6772    ~~ cen 6860    ~<_ cdom 6861
This theorem is referenced by:  mappwen  7739  pwcfsdom  8205  cfpwsdom  8206  rpnnen  12505  rexpen  12506  hauspwdom  17227
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-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-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-map 6774  df-en 6864  df-dom 6865
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