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Theorem mapdom1 7021
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 ) )
Dummy variable  x is distinct from all other variables.

Proof of Theorem mapdom1
StepHypRef Expression
1 reldom 6864 . . . . . . 7  |-  Rel  ~<_
21brrelex2i 4729 . . . . . 6  |-  ( A  ~<_  B  ->  B  e.  _V )
3 domeng 6871 . . . . . 6  |-  ( B  e.  _V  ->  ( A  ~<_  B  <->  E. x
( A  ~~  x  /\  x  C_  B ) ) )
42, 3syl 17 . . . . 5  |-  ( A  ~<_  B  ->  ( A  ~<_  B 
<->  E. x ( A 
~~  x  /\  x  C_  B ) ) )
54ibi 234 . . . 4  |-  ( A  ~<_  B  ->  E. x
( A  ~~  x  /\  x  C_  B ) )
65adantr 453 . . 3  |-  ( ( A  ~<_  B  /\  C  e.  _V )  ->  E. x
( A  ~~  x  /\  x  C_  B ) )
7 simpl 445 . . . . . . 7  |-  ( ( A  ~~  x  /\  x  C_  B )  ->  A  ~~  x )
8 enrefg 6888 . . . . . . . 8  |-  ( C  e.  _V  ->  C  ~~  C )
98adantl 454 . . . . . . 7  |-  ( ( A  ~<_  B  /\  C  e.  _V )  ->  C  ~~  C )
10 mapen 7020 . . . . . . 7  |-  ( ( A  ~~  x  /\  C  ~~  C )  -> 
( A  ^m  C
)  ~~  ( x  ^m  C ) )
117, 9, 10syl2anr 466 . . . . . 6  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( A  ~~  x  /\  x  C_  B ) )  ->  ( A  ^m  C )  ~~  (
x  ^m  C )
)
12 ovex 5844 . . . . . . 7  |-  ( B  ^m  C )  e. 
_V
132ad2antrr 708 . . . . . . . 8  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( A  ~~  x  /\  x  C_  B ) )  ->  B  e.  _V )
14 simprr 735 . . . . . . . 8  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( A  ~~  x  /\  x  C_  B ) )  ->  x  C_  B
)
15 mapss 6805 . . . . . . . 8  |-  ( ( B  e.  _V  /\  x  C_  B )  -> 
( x  ^m  C
)  C_  ( B  ^m  C ) )
1613, 14, 15syl2anc 644 . . . . . . 7  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( A  ~~  x  /\  x  C_  B ) )  ->  ( x  ^m  C )  C_  ( B  ^m  C ) )
17 ssdomg 6902 . . . . . . 7  |-  ( ( B  ^m  C )  e.  _V  ->  (
( x  ^m  C
)  C_  ( B  ^m  C )  ->  (
x  ^m  C )  ~<_  ( B  ^m  C ) ) )
1812, 16, 17mpsyl 61 . . . . . 6  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( A  ~~  x  /\  x  C_  B ) )  ->  ( x  ^m  C )  ~<_  ( B  ^m  C ) )
19 endomtr 6914 . . . . . 6  |-  ( ( ( A  ^m  C
)  ~~  ( x  ^m  C )  /\  (
x  ^m  C )  ~<_  ( B  ^m  C ) )  ->  ( A  ^m  C )  ~<_  ( B  ^m  C ) )
2011, 18, 19syl2anc 644 . . . . 5  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( A  ~~  x  /\  x  C_  B ) )  ->  ( A  ^m  C )  ~<_  ( B  ^m  C ) )
2120ex 425 . . . 4  |-  ( ( A  ~<_  B  /\  C  e.  _V )  ->  (
( A  ~~  x  /\  x  C_  B )  ->  ( A  ^m  C )  ~<_  ( B  ^m  C ) ) )
2221exlimdv 1665 . . 3  |-  ( ( A  ~<_  B  /\  C  e.  _V )  ->  ( E. x ( A  ~~  x  /\  x  C_  B
)  ->  ( A  ^m  C )  ~<_  ( B  ^m  C ) ) )
236, 22mpd 16 . 2  |-  ( ( A  ~<_  B  /\  C  e.  _V )  ->  ( A  ^m  C )  ~<_  ( B  ^m  C ) )
24 elmapex 6786 . . . . . . 7  |-  ( x  e.  ( A  ^m  C )  ->  ( A  e.  _V  /\  C  e.  _V ) )
2524simprd 451 . . . . . 6  |-  ( x  e.  ( A  ^m  C )  ->  C  e.  _V )
2625con3i 129 . . . . 5  |-  ( -.  C  e.  _V  ->  -.  x  e.  ( A  ^m  C ) )
2726eq0rdv 3490 . . . 4  |-  ( -.  C  e.  _V  ->  ( A  ^m  C )  =  (/) )
2827adantl 454 . . 3  |-  ( ( A  ~<_  B  /\  -.  C  e.  _V )  ->  ( A  ^m  C
)  =  (/) )
29120dom 6986 . . 3  |-  (/)  ~<_  ( B  ^m  C )
3028, 29syl6eqbr 4061 . 2  |-  ( ( A  ~<_  B  /\  -.  C  e.  _V )  ->  ( A  ^m  C
)  ~<_  ( B  ^m  C ) )
3123, 30pm2.61dan 768 1  |-  ( A  ~<_  B  ->  ( A  ^m  C )  ~<_  ( B  ^m  C ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360   E.wex 1529    = wceq 1624    e. wcel 1685   _Vcvv 2789    C_ wss 3153   (/)c0 3456   class class class wbr 4024  (class class class)co 5819    ^m cmap 6767    ~~ cen 6855    ~<_ cdom 6856
This theorem is referenced by:  mappwen  7734  pwcfsdom  8200  cfpwsdom  8201  rpnnen  12499  rexpen  12500  hauspwdom  17221
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-map 6769  df-en 6859  df-dom 6860
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