MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mapdom2 Structured version   Unicode version

Theorem mapdom2 7280
Description: Order-preserving property of set exponentiation. Theorem 6L(d) of [Enderton] p. 149. (Contributed by NM, 23-Sep-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)
Assertion
Ref Expression
mapdom2  |-  ( ( A  ~<_  B  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  -> 
( C  ^m  A
)  ~<_  ( C  ^m  B ) )

Proof of Theorem mapdom2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simpr 449 . . . . . . . 8  |-  ( ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  /\  C  =  (/) )  ->  C  =  (/) )
21oveq1d 6098 . . . . . . 7  |-  ( ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  /\  C  =  (/) )  ->  ( C  ^m  A )  =  ( (/)  ^m  A ) )
3 simplr 733 . . . . . . . . . 10  |-  ( ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  /\  C  =  (/) )  ->  -.  ( A  =  (/)  /\  C  =  (/) ) )
4 idd 23 . . . . . . . . . . 11  |-  ( ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  /\  C  =  (/) )  ->  ( A  =  (/)  ->  A  =  (/) ) )
54, 1jctird 530 . . . . . . . . . 10  |-  ( ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  /\  C  =  (/) )  ->  ( A  =  (/)  ->  ( A  =  (/)  /\  C  =  (/) ) ) )
63, 5mtod 171 . . . . . . . . 9  |-  ( ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  /\  C  =  (/) )  ->  -.  A  =  (/) )
76neneqad 2676 . . . . . . . 8  |-  ( ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  /\  C  =  (/) )  ->  A  =/=  (/) )
8 map0b 7054 . . . . . . . 8  |-  ( A  =/=  (/)  ->  ( (/)  ^m  A
)  =  (/) )
97, 8syl 16 . . . . . . 7  |-  ( ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  /\  C  =  (/) )  ->  ( (/) 
^m  A )  =  (/) )
102, 9eqtrd 2470 . . . . . 6  |-  ( ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  /\  C  =  (/) )  ->  ( C  ^m  A )  =  (/) )
11 ovex 6108 . . . . . . 7  |-  ( C  ^m  B )  e. 
_V
12110dom 7239 . . . . . 6  |-  (/)  ~<_  ( C  ^m  B )
1310, 12syl6eqbr 4251 . . . . 5  |-  ( ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  /\  C  =  (/) )  ->  ( C  ^m  A )  ~<_  ( C  ^m  B ) )
14 simpll 732 . . . . . . . 8  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  C  =/=  (/) )  ->  A  ~<_  B )
15 reldom 7117 . . . . . . . . . . 11  |-  Rel  ~<_
1615brrelex2i 4921 . . . . . . . . . 10  |-  ( A  ~<_  B  ->  B  e.  _V )
1716ad2antrr 708 . . . . . . . . 9  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  C  =/=  (/) )  ->  B  e.  _V )
18 domeng 7124 . . . . . . . . 9  |-  ( B  e.  _V  ->  ( A  ~<_  B  <->  E. x
( A  ~~  x  /\  x  C_  B ) ) )
1917, 18syl 16 . . . . . . . 8  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  C  =/=  (/) )  -> 
( A  ~<_  B  <->  E. x
( A  ~~  x  /\  x  C_  B ) ) )
2014, 19mpbid 203 . . . . . . 7  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  C  =/=  (/) )  ->  E. x ( A  ~~  x  /\  x  C_  B
) )
21 enrefg 7141 . . . . . . . . . . . 12  |-  ( C  e.  _V  ->  C  ~~  C )
2221ad2antlr 709 . . . . . . . . . . 11  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( C  =/=  (/)  /\  ( A  ~~  x  /\  x  C_  B ) ) )  ->  C  ~~  C
)
23 simprrl 742 . . . . . . . . . . 11  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( C  =/=  (/)  /\  ( A  ~~  x  /\  x  C_  B ) ) )  ->  A  ~~  x
)
24 mapen 7273 . . . . . . . . . . 11  |-  ( ( C  ~~  C  /\  A  ~~  x )  -> 
( C  ^m  A
)  ~~  ( C  ^m  x ) )
2522, 23, 24syl2anc 644 . . . . . . . . . 10  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( C  =/=  (/)  /\  ( A  ~~  x  /\  x  C_  B ) ) )  ->  ( C  ^m  A )  ~~  ( C  ^m  x ) )
26 ovex 6108 . . . . . . . . . . . . 13  |-  ( C  ^m  x )  e. 
_V
2726a1i 11 . . . . . . . . . . . 12  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( C  =/=  (/)  /\  ( A  ~~  x  /\  x  C_  B ) ) )  ->  ( C  ^m  x )  e.  _V )
28 ovex 6108 . . . . . . . . . . . . 13  |-  ( C  ^m  ( B  \  x ) )  e. 
_V
2928a1i 11 . . . . . . . . . . . 12  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( C  =/=  (/)  /\  ( A  ~~  x  /\  x  C_  B ) ) )  ->  ( C  ^m  ( B  \  x
) )  e.  _V )
30 simprl 734 . . . . . . . . . . . . 13  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( C  =/=  (/)  /\  ( A  ~~  x  /\  x  C_  B ) ) )  ->  C  =/=  (/) )
31 simplr 733 . . . . . . . . . . . . . 14  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( C  =/=  (/)  /\  ( A  ~~  x  /\  x  C_  B ) ) )  ->  C  e.  _V )
3216ad2antrr 708 . . . . . . . . . . . . . . 15  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( C  =/=  (/)  /\  ( A  ~~  x  /\  x  C_  B ) ) )  ->  B  e.  _V )
33 difexg 4353 . . . . . . . . . . . . . . 15  |-  ( B  e.  _V  ->  ( B  \  x )  e. 
_V )
3432, 33syl 16 . . . . . . . . . . . . . 14  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( C  =/=  (/)  /\  ( A  ~~  x  /\  x  C_  B ) ) )  ->  ( B  \  x )  e.  _V )
35 map0g 7055 . . . . . . . . . . . . . . . 16  |-  ( ( C  e.  _V  /\  ( B  \  x
)  e.  _V )  ->  ( ( C  ^m  ( B  \  x
) )  =  (/)  <->  ( C  =  (/)  /\  ( B  \  x )  =/=  (/) ) ) )
36 simpl 445 . . . . . . . . . . . . . . . 16  |-  ( ( C  =  (/)  /\  ( B  \  x )  =/=  (/) )  ->  C  =  (/) )
3735, 36syl6bi 221 . . . . . . . . . . . . . . 15  |-  ( ( C  e.  _V  /\  ( B  \  x
)  e.  _V )  ->  ( ( C  ^m  ( B  \  x
) )  =  (/)  ->  C  =  (/) ) )
3837necon3d 2641 . . . . . . . . . . . . . 14  |-  ( ( C  e.  _V  /\  ( B  \  x
)  e.  _V )  ->  ( C  =/=  (/)  ->  ( C  ^m  ( B  \  x ) )  =/=  (/) ) )
3931, 34, 38syl2anc 644 . . . . . . . . . . . . 13  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( C  =/=  (/)  /\  ( A  ~~  x  /\  x  C_  B ) ) )  ->  ( C  =/=  (/)  ->  ( C  ^m  ( B  \  x
) )  =/=  (/) ) )
4030, 39mpd 15 . . . . . . . . . . . 12  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( C  =/=  (/)  /\  ( A  ~~  x  /\  x  C_  B ) ) )  ->  ( C  ^m  ( B  \  x
) )  =/=  (/) )
41 xpdom3 7208 . . . . . . . . . . . 12  |-  ( ( ( C  ^m  x
)  e.  _V  /\  ( C  ^m  ( B  \  x ) )  e.  _V  /\  ( C  ^m  ( B  \  x ) )  =/=  (/) )  ->  ( C  ^m  x )  ~<_  ( ( C  ^m  x
)  X.  ( C  ^m  ( B  \  x ) ) ) )
4227, 29, 40, 41syl3anc 1185 . . . . . . . . . . 11  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( C  =/=  (/)  /\  ( A  ~~  x  /\  x  C_  B ) ) )  ->  ( C  ^m  x )  ~<_  ( ( C  ^m  x )  X.  ( C  ^m  ( B  \  x
) ) ) )
43 vex 2961 . . . . . . . . . . . . . . 15  |-  x  e. 
_V
4443a1i 11 . . . . . . . . . . . . . 14  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( C  =/=  (/)  /\  ( A  ~~  x  /\  x  C_  B ) ) )  ->  x  e.  _V )
45 disjdif 3702 . . . . . . . . . . . . . . 15  |-  ( x  i^i  ( B  \  x ) )  =  (/)
4645a1i 11 . . . . . . . . . . . . . 14  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( C  =/=  (/)  /\  ( A  ~~  x  /\  x  C_  B ) ) )  ->  ( x  i^i  ( B  \  x
) )  =  (/) )
47 mapunen 7278 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  _V  /\  ( B  \  x
)  e.  _V  /\  C  e.  _V )  /\  ( x  i^i  ( B  \  x ) )  =  (/) )  ->  ( C  ^m  ( x  u.  ( B  \  x
) ) )  ~~  ( ( C  ^m  x )  X.  ( C  ^m  ( B  \  x ) ) ) )
4844, 34, 31, 46, 47syl31anc 1188 . . . . . . . . . . . . 13  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( C  =/=  (/)  /\  ( A  ~~  x  /\  x  C_  B ) ) )  ->  ( C  ^m  ( x  u.  ( B  \  x ) ) )  ~~  ( ( C  ^m  x )  X.  ( C  ^m  ( B  \  x
) ) ) )
4948ensymd 7160 . . . . . . . . . . . 12  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( C  =/=  (/)  /\  ( A  ~~  x  /\  x  C_  B ) ) )  ->  ( ( C  ^m  x )  X.  ( C  ^m  ( B  \  x ) ) )  ~~  ( C  ^m  ( x  u.  ( B  \  x
) ) ) )
50 simprrr 743 . . . . . . . . . . . . . 14  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( C  =/=  (/)  /\  ( A  ~~  x  /\  x  C_  B ) ) )  ->  x  C_  B
)
51 undif 3710 . . . . . . . . . . . . . 14  |-  ( x 
C_  B  <->  ( x  u.  ( B  \  x
) )  =  B )
5250, 51sylib 190 . . . . . . . . . . . . 13  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( C  =/=  (/)  /\  ( A  ~~  x  /\  x  C_  B ) ) )  ->  ( x  u.  ( B  \  x
) )  =  B )
5352oveq2d 6099 . . . . . . . . . . . 12  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( C  =/=  (/)  /\  ( A  ~~  x  /\  x  C_  B ) ) )  ->  ( C  ^m  ( x  u.  ( B  \  x ) ) )  =  ( C  ^m  B ) )
5449, 53breqtrd 4238 . . . . . . . . . . 11  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( C  =/=  (/)  /\  ( A  ~~  x  /\  x  C_  B ) ) )  ->  ( ( C  ^m  x )  X.  ( C  ^m  ( B  \  x ) ) )  ~~  ( C  ^m  B ) )
55 domentr 7168 . . . . . . . . . . 11  |-  ( ( ( C  ^m  x
)  ~<_  ( ( C  ^m  x )  X.  ( C  ^m  ( B  \  x ) ) )  /\  ( ( C  ^m  x )  X.  ( C  ^m  ( B  \  x
) ) )  ~~  ( C  ^m  B ) )  ->  ( C  ^m  x )  ~<_  ( C  ^m  B ) )
5642, 54, 55syl2anc 644 . . . . . . . . . 10  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( C  =/=  (/)  /\  ( A  ~~  x  /\  x  C_  B ) ) )  ->  ( C  ^m  x )  ~<_  ( C  ^m  B ) )
57 endomtr 7167 . . . . . . . . . 10  |-  ( ( ( C  ^m  A
)  ~~  ( C  ^m  x )  /\  ( C  ^m  x )  ~<_  ( C  ^m  B ) )  ->  ( C  ^m  A )  ~<_  ( C  ^m  B ) )
5825, 56, 57syl2anc 644 . . . . . . . . 9  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( C  =/=  (/)  /\  ( A  ~~  x  /\  x  C_  B ) ) )  ->  ( C  ^m  A )  ~<_  ( C  ^m  B ) )
5958expr 600 . . . . . . . 8  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  C  =/=  (/) )  -> 
( ( A  ~~  x  /\  x  C_  B
)  ->  ( C  ^m  A )  ~<_  ( C  ^m  B ) ) )
6059exlimdv 1647 . . . . . . 7  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  C  =/=  (/) )  -> 
( E. x ( A  ~~  x  /\  x  C_  B )  -> 
( C  ^m  A
)  ~<_  ( C  ^m  B ) ) )
6120, 60mpd 15 . . . . . 6  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  C  =/=  (/) )  -> 
( C  ^m  A
)  ~<_  ( C  ^m  B ) )
6261adantlr 697 . . . . 5  |-  ( ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  /\  C  =/=  (/) )  ->  ( C  ^m  A )  ~<_  ( C  ^m  B ) )
6313, 62pm2.61dane 2684 . . . 4  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  ->  ( C  ^m  A )  ~<_  ( C  ^m  B ) )
6463an32s 781 . . 3  |-  ( ( ( A  ~<_  B  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  /\  C  e.  _V )  ->  ( C  ^m  A )  ~<_  ( C  ^m  B ) )
6564ex 425 . 2  |-  ( ( A  ~<_  B  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  -> 
( C  e.  _V  ->  ( C  ^m  A
)  ~<_  ( C  ^m  B ) ) )
66 reldmmap 7029 . . . 4  |-  Rel  dom  ^m
6766ovprc1 6111 . . 3  |-  ( -.  C  e.  _V  ->  ( C  ^m  A )  =  (/) )
6867, 12syl6eqbr 4251 . 2  |-  ( -.  C  e.  _V  ->  ( C  ^m  A )  ~<_  ( C  ^m  B
) )
6965, 68pm2.61d1 154 1  |-  ( ( A  ~<_  B  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  -> 
( C  ^m  A
)  ~<_  ( C  ^m  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360   E.wex 1551    = wceq 1653    e. wcel 1726    =/= wne 2601   _Vcvv 2958    \ cdif 3319    u. cun 3320    i^i cin 3321    C_ wss 3322   (/)c0 3630   class class class wbr 4214    X. cxp 4878  (class class class)co 6083    ^m cmap 7020    ~~ cen 7108    ~<_ cdom 7109
This theorem is referenced by:  mapdom3  7281  cfpwsdom  8461  hauspwdom  17566
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-er 6907  df-map 7022  df-en 7112  df-dom 7113
  Copyright terms: Public domain W3C validator