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Theorem mapdom2 7269
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 448 . . . . . . . 8  |-  ( ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  /\  C  =  (/) )  ->  C  =  (/) )
21oveq1d 6087 . . . . . . 7  |-  ( ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  /\  C  =  (/) )  ->  ( C  ^m  A )  =  ( (/)  ^m  A ) )
3 simplr 732 . . . . . . . . . 10  |-  ( ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  /\  C  =  (/) )  ->  -.  ( A  =  (/)  /\  C  =  (/) ) )
4 idd 22 . . . . . . . . . . 11  |-  ( ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  /\  C  =  (/) )  ->  ( A  =  (/)  ->  A  =  (/) ) )
54, 1jctird 529 . . . . . . . . . 10  |-  ( ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  /\  C  =  (/) )  ->  ( A  =  (/)  ->  ( A  =  (/)  /\  C  =  (/) ) ) )
63, 5mtod 170 . . . . . . . . 9  |-  ( ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  /\  C  =  (/) )  ->  -.  A  =  (/) )
76neneqad 2668 . . . . . . . 8  |-  ( ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  /\  C  =  (/) )  ->  A  =/=  (/) )
8 map0b 7043 . . . . . . . 8  |-  ( A  =/=  (/)  ->  ( (/)  ^m  A
)  =  (/) )
97, 8syl 16 . . . . . . 7  |-  ( ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  /\  C  =  (/) )  ->  ( (/) 
^m  A )  =  (/) )
102, 9eqtrd 2467 . . . . . 6  |-  ( ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  /\  C  =  (/) )  ->  ( C  ^m  A )  =  (/) )
11 ovex 6097 . . . . . . 7  |-  ( C  ^m  B )  e. 
_V
12110dom 7228 . . . . . 6  |-  (/)  ~<_  ( C  ^m  B )
1310, 12syl6eqbr 4241 . . . . 5  |-  ( ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  /\  C  =  (/) )  ->  ( C  ^m  A )  ~<_  ( C  ^m  B ) )
14 simpll 731 . . . . . . . 8  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  C  =/=  (/) )  ->  A  ~<_  B )
15 reldom 7106 . . . . . . . . . . 11  |-  Rel  ~<_
1615brrelex2i 4910 . . . . . . . . . 10  |-  ( A  ~<_  B  ->  B  e.  _V )
1716ad2antrr 707 . . . . . . . . 9  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  C  =/=  (/) )  ->  B  e.  _V )
18 domeng 7113 . . . . . . . . 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 202 . . . . . . 7  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  C  =/=  (/) )  ->  E. x ( A  ~~  x  /\  x  C_  B
) )
21 enrefg 7130 . . . . . . . . . . . 12  |-  ( C  e.  _V  ->  C  ~~  C )
2221ad2antlr 708 . . . . . . . . . . 11  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( C  =/=  (/)  /\  ( A  ~~  x  /\  x  C_  B ) ) )  ->  C  ~~  C
)
23 simprrl 741 . . . . . . . . . . 11  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( C  =/=  (/)  /\  ( A  ~~  x  /\  x  C_  B ) ) )  ->  A  ~~  x
)
24 mapen 7262 . . . . . . . . . . 11  |-  ( ( C  ~~  C  /\  A  ~~  x )  -> 
( C  ^m  A
)  ~~  ( C  ^m  x ) )
2522, 23, 24syl2anc 643 . . . . . . . . . 10  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( C  =/=  (/)  /\  ( A  ~~  x  /\  x  C_  B ) ) )  ->  ( C  ^m  A )  ~~  ( C  ^m  x ) )
26 ovex 6097 . . . . . . . . . . . . 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 6097 . . . . . . . . . . . . 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 733 . . . . . . . . . . . . 13  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( C  =/=  (/)  /\  ( A  ~~  x  /\  x  C_  B ) ) )  ->  C  =/=  (/) )
31 simplr 732 . . . . . . . . . . . . . 14  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( C  =/=  (/)  /\  ( A  ~~  x  /\  x  C_  B ) ) )  ->  C  e.  _V )
3216ad2antrr 707 . . . . . . . . . . . . . . 15  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( C  =/=  (/)  /\  ( A  ~~  x  /\  x  C_  B ) ) )  ->  B  e.  _V )
33 difexg 4343 . . . . . . . . . . . . . . 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 7044 . . . . . . . . . . . . . . . 16  |-  ( ( C  e.  _V  /\  ( B  \  x
)  e.  _V )  ->  ( ( C  ^m  ( B  \  x
) )  =  (/)  <->  ( C  =  (/)  /\  ( B  \  x )  =/=  (/) ) ) )
36 simpl 444 . . . . . . . . . . . . . . . 16  |-  ( ( C  =  (/)  /\  ( B  \  x )  =/=  (/) )  ->  C  =  (/) )
3735, 36syl6bi 220 . . . . . . . . . . . . . . 15  |-  ( ( C  e.  _V  /\  ( B  \  x
)  e.  _V )  ->  ( ( C  ^m  ( B  \  x
) )  =  (/)  ->  C  =  (/) ) )
3837necon3d 2636 . . . . . . . . . . . . . 14  |-  ( ( C  e.  _V  /\  ( B  \  x
)  e.  _V )  ->  ( C  =/=  (/)  ->  ( C  ^m  ( B  \  x ) )  =/=  (/) ) )
3931, 34, 38syl2anc 643 . . . . . . . . . . . . 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 7197 . . . . . . . . . . . 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 1184 . . . . . . . . . . 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 2951 . . . . . . . . . . . . . . 15  |-  x  e. 
_V
4443a1i 11 . . . . . . . . . . . . . 14  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( C  =/=  (/)  /\  ( A  ~~  x  /\  x  C_  B ) ) )  ->  x  e.  _V )
45 disjdif 3692 . . . . . . . . . . . . . . 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 7267 . . . . . . . . . . . . . 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 1187 . . . . . . . . . . . . 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 7149 . . . . . . . . . . . 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 742 . . . . . . . . . . . . . 14  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( C  =/=  (/)  /\  ( A  ~~  x  /\  x  C_  B ) ) )  ->  x  C_  B
)
51 undif 3700 . . . . . . . . . . . . . 14  |-  ( x 
C_  B  <->  ( x  u.  ( B  \  x
) )  =  B )
5250, 51sylib 189 . . . . . . . . . . . . 13  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( C  =/=  (/)  /\  ( A  ~~  x  /\  x  C_  B ) ) )  ->  ( x  u.  ( B  \  x
) )  =  B )
5352oveq2d 6088 . . . . . . . . . . . 12  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( C  =/=  (/)  /\  ( A  ~~  x  /\  x  C_  B ) ) )  ->  ( C  ^m  ( x  u.  ( B  \  x ) ) )  =  ( C  ^m  B ) )
5449, 53breqtrd 4228 . . . . . . . . . . 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 7157 . . . . . . . . . . 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 643 . . . . . . . . . 10  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( C  =/=  (/)  /\  ( A  ~~  x  /\  x  C_  B ) ) )  ->  ( C  ^m  x )  ~<_  ( C  ^m  B ) )
57 endomtr 7156 . . . . . . . . . 10  |-  ( ( ( C  ^m  A
)  ~~  ( C  ^m  x )  /\  ( C  ^m  x )  ~<_  ( C  ^m  B ) )  ->  ( C  ^m  A )  ~<_  ( C  ^m  B ) )
5825, 56, 57syl2anc 643 . . . . . . . . 9  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( C  =/=  (/)  /\  ( A  ~~  x  /\  x  C_  B ) ) )  ->  ( C  ^m  A )  ~<_  ( C  ^m  B ) )
5958expr 599 . . . . . . . 8  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  C  =/=  (/) )  -> 
( ( A  ~~  x  /\  x  C_  B
)  ->  ( C  ^m  A )  ~<_  ( C  ^m  B ) ) )
6059exlimdv 1646 . . . . . . 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 696 . . . . 5  |-  ( ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  /\  C  =/=  (/) )  ->  ( C  ^m  A )  ~<_  ( C  ^m  B ) )
6313, 62pm2.61dane 2676 . . . 4  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  ->  ( C  ^m  A )  ~<_  ( C  ^m  B ) )
6463an32s 780 . . 3  |-  ( ( ( A  ~<_  B  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  /\  C  e.  _V )  ->  ( C  ^m  A )  ~<_  ( C  ^m  B ) )
6564ex 424 . 2  |-  ( ( A  ~<_  B  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  -> 
( C  e.  _V  ->  ( C  ^m  A
)  ~<_  ( C  ^m  B ) ) )
66 reldmmap 7018 . . . 4  |-  Rel  dom  ^m
6766ovprc1 6100 . . 3  |-  ( -.  C  e.  _V  ->  ( C  ^m  A )  =  (/) )
6867, 12syl6eqbr 4241 . 2  |-  ( -.  C  e.  _V  ->  ( C  ^m  A )  ~<_  ( C  ^m  B
) )
6965, 68pm2.61d1 153 1  |-  ( ( A  ~<_  B  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  -> 
( C  ^m  A
)  ~<_  ( C  ^m  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725    =/= wne 2598   _Vcvv 2948    \ cdif 3309    u. cun 3310    i^i cin 3311    C_ wss 3312   (/)c0 3620   class class class wbr 4204    X. cxp 4867  (class class class)co 6072    ^m cmap 7009    ~~ cen 7097    ~<_ cdom 7098
This theorem is referenced by:  mapdom3  7270  cfpwsdom  8448  hauspwdom  17552
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-er 6896  df-map 7011  df-en 7101  df-dom 7102
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