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Theorem mapdom1 7042
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 6885 . . . . . . 7  |-  Rel  ~<_
21brrelex2i 4746 . . . . . 6  |-  ( A  ~<_  B  ->  B  e.  _V )
3 domeng 6892 . . . . . 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 6909 . . . . . . . 8  |-  ( C  e.  _V  ->  C  ~~  C )
98adantl 452 . . . . . . 7  |-  ( ( A  ~<_  B  /\  C  e.  _V )  ->  C  ~~  C )
10 mapen 7041 . . . . . . 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 5899 . . . . . . 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 6826 . . . . . . . 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 6923 . . . . . . 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 6935 . . . . . 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 1626 . . 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 6807 . . . . . . 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 3502 . . . 4  |-  ( -.  C  e.  _V  ->  ( A  ^m  C )  =  (/) )
2827adantl 452 . . 3  |-  ( ( A  ~<_  B  /\  -.  C  e.  _V )  ->  ( A  ^m  C
)  =  (/) )
29120dom 7007 . . 3  |-  (/)  ~<_  ( B  ^m  C )
3028, 29syl6eqbr 4076 . 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 1531    = wceq 1632    e. wcel 1696   _Vcvv 2801    C_ wss 3165   (/)c0 3468   class class class wbr 4039  (class class class)co 5874    ^m cmap 6788    ~~ cen 6876    ~<_ cdom 6877
This theorem is referenced by:  mappwen  7755  pwcfsdom  8221  cfpwsdom  8222  rpnnen  12521  rexpen  12522  hauspwdom  17243
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-map 6790  df-en 6880  df-dom 6881
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