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Theorem mapdom1 7682
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 7522 . . . . . . 7  |-  Rel  ~<_
21brrelex2i 5040 . . . . . 6  |-  ( A  ~<_  B  ->  B  e.  _V )
3 domeng 7530 . . . . . 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 241 . . . 4  |-  ( A  ~<_  B  ->  E. x
( A  ~~  x  /\  x  C_  B ) )
65adantr 465 . . 3  |-  ( ( A  ~<_  B  /\  C  e.  _V )  ->  E. x
( A  ~~  x  /\  x  C_  B ) )
7 simpl 457 . . . . 5  |-  ( ( A  ~~  x  /\  x  C_  B )  ->  A  ~~  x )
8 enrefg 7547 . . . . . 6  |-  ( C  e.  _V  ->  C  ~~  C )
98adantl 466 . . . . 5  |-  ( ( A  ~<_  B  /\  C  e.  _V )  ->  C  ~~  C )
10 mapen 7681 . . . . 5  |-  ( ( A  ~~  x  /\  C  ~~  C )  -> 
( A  ^m  C
)  ~~  ( x  ^m  C ) )
117, 9, 10syl2anr 478 . . . 4  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( A  ~~  x  /\  x  C_  B ) )  ->  ( A  ^m  C )  ~~  (
x  ^m  C )
)
12 ovex 6308 . . . . 5  |-  ( B  ^m  C )  e. 
_V
132ad2antrr 725 . . . . . 6  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( A  ~~  x  /\  x  C_  B ) )  ->  B  e.  _V )
14 simprr 756 . . . . . 6  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( A  ~~  x  /\  x  C_  B ) )  ->  x  C_  B
)
15 mapss 7461 . . . . . 6  |-  ( ( B  e.  _V  /\  x  C_  B )  -> 
( x  ^m  C
)  C_  ( B  ^m  C ) )
1613, 14, 15syl2anc 661 . . . . 5  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( A  ~~  x  /\  x  C_  B ) )  ->  ( x  ^m  C )  C_  ( B  ^m  C ) )
17 ssdomg 7561 . . . . 5  |-  ( ( B  ^m  C )  e.  _V  ->  (
( x  ^m  C
)  C_  ( B  ^m  C )  ->  (
x  ^m  C )  ~<_  ( B  ^m  C ) ) )
1812, 16, 17mpsyl 63 . . . 4  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( A  ~~  x  /\  x  C_  B ) )  ->  ( x  ^m  C )  ~<_  ( B  ^m  C ) )
19 endomtr 7573 . . . 4  |-  ( ( ( A  ^m  C
)  ~~  ( x  ^m  C )  /\  (
x  ^m  C )  ~<_  ( B  ^m  C ) )  ->  ( A  ^m  C )  ~<_  ( B  ^m  C ) )
2011, 18, 19syl2anc 661 . . 3  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( A  ~~  x  /\  x  C_  B ) )  ->  ( A  ^m  C )  ~<_  ( B  ^m  C ) )
216, 20exlimddv 1702 . 2  |-  ( ( A  ~<_  B  /\  C  e.  _V )  ->  ( A  ^m  C )  ~<_  ( B  ^m  C ) )
22 elmapex 7439 . . . . . . 7  |-  ( x  e.  ( A  ^m  C )  ->  ( A  e.  _V  /\  C  e.  _V ) )
2322simprd 463 . . . . . 6  |-  ( x  e.  ( A  ^m  C )  ->  C  e.  _V )
2423con3i 135 . . . . 5  |-  ( -.  C  e.  _V  ->  -.  x  e.  ( A  ^m  C ) )
2524eq0rdv 3820 . . . 4  |-  ( -.  C  e.  _V  ->  ( A  ^m  C )  =  (/) )
2625adantl 466 . . 3  |-  ( ( A  ~<_  B  /\  -.  C  e.  _V )  ->  ( A  ^m  C
)  =  (/) )
27120dom 7647 . . 3  |-  (/)  ~<_  ( B  ^m  C )
2826, 27syl6eqbr 4484 . 2  |-  ( ( A  ~<_  B  /\  -.  C  e.  _V )  ->  ( A  ^m  C
)  ~<_  ( B  ^m  C ) )
2921, 28pm2.61dan 789 1  |-  ( A  ~<_  B  ->  ( A  ^m  C )  ~<_  ( B  ^m  C ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767   _Vcvv 3113    C_ wss 3476   (/)c0 3785   class class class wbr 4447  (class class class)co 6283    ^m cmap 7420    ~~ cen 7513    ~<_ cdom 7514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-1st 6784  df-2nd 6785  df-map 7422  df-en 7517  df-dom 7518
This theorem is referenced by:  mappwen  8492  pwcfsdom  8957  cfpwsdom  8958  rpnnen  13820  rexpen  13821  hauspwdom  19784
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