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Theorem mapdom2 7707
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 461 . . . . . . . 8  |-  ( ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  /\  C  =  (/) )  ->  C  =  (/) )
21oveq1d 6311 . . . . . . 7  |-  ( ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  /\  C  =  (/) )  ->  ( C  ^m  A )  =  ( (/)  ^m  A ) )
3 simplr 755 . . . . . . . . . 10  |-  ( ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  /\  C  =  (/) )  ->  -.  ( A  =  (/)  /\  C  =  (/) ) )
4 idd 24 . . . . . . . . . . 11  |-  ( ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  /\  C  =  (/) )  ->  ( A  =  (/)  ->  A  =  (/) ) )
54, 1jctird 544 . . . . . . . . . 10  |-  ( ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  /\  C  =  (/) )  ->  ( A  =  (/)  ->  ( A  =  (/)  /\  C  =  (/) ) ) )
63, 5mtod 177 . . . . . . . . 9  |-  ( ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  /\  C  =  (/) )  ->  -.  A  =  (/) )
76neqned 2660 . . . . . . . 8  |-  ( ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  /\  C  =  (/) )  ->  A  =/=  (/) )
8 map0b 7476 . . . . . . . 8  |-  ( A  =/=  (/)  ->  ( (/)  ^m  A
)  =  (/) )
97, 8syl 16 . . . . . . 7  |-  ( ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  /\  C  =  (/) )  ->  ( (/) 
^m  A )  =  (/) )
102, 9eqtrd 2498 . . . . . 6  |-  ( ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  /\  C  =  (/) )  ->  ( C  ^m  A )  =  (/) )
11 ovex 6324 . . . . . . 7  |-  ( C  ^m  B )  e. 
_V
12110dom 7666 . . . . . 6  |-  (/)  ~<_  ( C  ^m  B )
1310, 12syl6eqbr 4493 . . . . 5  |-  ( ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  /\  C  =  (/) )  ->  ( C  ^m  A )  ~<_  ( C  ^m  B ) )
14 simpll 753 . . . . . . . 8  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  C  =/=  (/) )  ->  A  ~<_  B )
15 reldom 7541 . . . . . . . . . . 11  |-  Rel  ~<_
1615brrelex2i 5050 . . . . . . . . . 10  |-  ( A  ~<_  B  ->  B  e.  _V )
1716ad2antrr 725 . . . . . . . . 9  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  C  =/=  (/) )  ->  B  e.  _V )
18 domeng 7549 . . . . . . . . 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 210 . . . . . . 7  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  C  =/=  (/) )  ->  E. x ( A  ~~  x  /\  x  C_  B
) )
21 enrefg 7566 . . . . . . . . . . . 12  |-  ( C  e.  _V  ->  C  ~~  C )
2221ad2antlr 726 . . . . . . . . . . 11  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( C  =/=  (/)  /\  ( A  ~~  x  /\  x  C_  B ) ) )  ->  C  ~~  C
)
23 simprrl 765 . . . . . . . . . . 11  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( C  =/=  (/)  /\  ( A  ~~  x  /\  x  C_  B ) ) )  ->  A  ~~  x
)
24 mapen 7700 . . . . . . . . . . 11  |-  ( ( C  ~~  C  /\  A  ~~  x )  -> 
( C  ^m  A
)  ~~  ( C  ^m  x ) )
2522, 23, 24syl2anc 661 . . . . . . . . . 10  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( C  =/=  (/)  /\  ( A  ~~  x  /\  x  C_  B ) ) )  ->  ( C  ^m  A )  ~~  ( C  ^m  x ) )
26 ovex 6324 . . . . . . . . . . . . 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 6324 . . . . . . . . . . . . 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 756 . . . . . . . . . . . . 13  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( C  =/=  (/)  /\  ( A  ~~  x  /\  x  C_  B ) ) )  ->  C  =/=  (/) )
31 simplr 755 . . . . . . . . . . . . . 14  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( C  =/=  (/)  /\  ( A  ~~  x  /\  x  C_  B ) ) )  ->  C  e.  _V )
3216ad2antrr 725 . . . . . . . . . . . . . . 15  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( C  =/=  (/)  /\  ( A  ~~  x  /\  x  C_  B ) ) )  ->  B  e.  _V )
33 difexg 4604 . . . . . . . . . . . . . . 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 7477 . . . . . . . . . . . . . . . 16  |-  ( ( C  e.  _V  /\  ( B  \  x
)  e.  _V )  ->  ( ( C  ^m  ( B  \  x
) )  =  (/)  <->  ( C  =  (/)  /\  ( B  \  x )  =/=  (/) ) ) )
36 simpl 457 . . . . . . . . . . . . . . . 16  |-  ( ( C  =  (/)  /\  ( B  \  x )  =/=  (/) )  ->  C  =  (/) )
3735, 36syl6bi 228 . . . . . . . . . . . . . . 15  |-  ( ( C  e.  _V  /\  ( B  \  x
)  e.  _V )  ->  ( ( C  ^m  ( B  \  x
) )  =  (/)  ->  C  =  (/) ) )
3837necon3d 2681 . . . . . . . . . . . . . 14  |-  ( ( C  e.  _V  /\  ( B  \  x
)  e.  _V )  ->  ( C  =/=  (/)  ->  ( C  ^m  ( B  \  x ) )  =/=  (/) ) )
3931, 34, 38syl2anc 661 . . . . . . . . . . . . 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 7634 . . . . . . . . . . . 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 1228 . . . . . . . . . . 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 3112 . . . . . . . . . . . . . . 15  |-  x  e. 
_V
4443a1i 11 . . . . . . . . . . . . . 14  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( C  =/=  (/)  /\  ( A  ~~  x  /\  x  C_  B ) ) )  ->  x  e.  _V )
45 disjdif 3903 . . . . . . . . . . . . . . 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 7705 . . . . . . . . . . . . . 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 1231 . . . . . . . . . . . . 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 7585 . . . . . . . . . . . 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 766 . . . . . . . . . . . . . 14  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( C  =/=  (/)  /\  ( A  ~~  x  /\  x  C_  B ) ) )  ->  x  C_  B
)
51 undif 3911 . . . . . . . . . . . . . 14  |-  ( x 
C_  B  <->  ( x  u.  ( B  \  x
) )  =  B )
5250, 51sylib 196 . . . . . . . . . . . . 13  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( C  =/=  (/)  /\  ( A  ~~  x  /\  x  C_  B ) ) )  ->  ( x  u.  ( B  \  x
) )  =  B )
5352oveq2d 6312 . . . . . . . . . . . 12  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( C  =/=  (/)  /\  ( A  ~~  x  /\  x  C_  B ) ) )  ->  ( C  ^m  ( x  u.  ( B  \  x ) ) )  =  ( C  ^m  B ) )
5449, 53breqtrd 4480 . . . . . . . . . . 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 7593 . . . . . . . . . . 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 661 . . . . . . . . . 10  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( C  =/=  (/)  /\  ( A  ~~  x  /\  x  C_  B ) ) )  ->  ( C  ^m  x )  ~<_  ( C  ^m  B ) )
57 endomtr 7592 . . . . . . . . . 10  |-  ( ( ( C  ^m  A
)  ~~  ( C  ^m  x )  /\  ( C  ^m  x )  ~<_  ( C  ^m  B ) )  ->  ( C  ^m  A )  ~<_  ( C  ^m  B ) )
5825, 56, 57syl2anc 661 . . . . . . . . 9  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( C  =/=  (/)  /\  ( A  ~~  x  /\  x  C_  B ) ) )  ->  ( C  ^m  A )  ~<_  ( C  ^m  B ) )
5958expr 615 . . . . . . . 8  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  C  =/=  (/) )  -> 
( ( A  ~~  x  /\  x  C_  B
)  ->  ( C  ^m  A )  ~<_  ( C  ^m  B ) ) )
6059exlimdv 1725 . . . . . . 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 714 . . . . 5  |-  ( ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  /\  C  =/=  (/) )  ->  ( C  ^m  A )  ~<_  ( C  ^m  B ) )
6313, 62pm2.61dane 2775 . . . 4  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  ->  ( C  ^m  A )  ~<_  ( C  ^m  B ) )
6463an32s 804 . . 3  |-  ( ( ( A  ~<_  B  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  /\  C  e.  _V )  ->  ( C  ^m  A )  ~<_  ( C  ^m  B ) )
6564ex 434 . 2  |-  ( ( A  ~<_  B  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  -> 
( C  e.  _V  ->  ( C  ^m  A
)  ~<_  ( C  ^m  B ) ) )
66 reldmmap 7447 . . . 4  |-  Rel  dom  ^m
6766ovprc1 6327 . . 3  |-  ( -.  C  e.  _V  ->  ( C  ^m  A )  =  (/) )
6867, 12syl6eqbr 4493 . 2  |-  ( -.  C  e.  _V  ->  ( C  ^m  A )  ~<_  ( C  ^m  B
) )
6965, 68pm2.61d1 159 1  |-  ( ( A  ~<_  B  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  -> 
( C  ^m  A
)  ~<_  ( C  ^m  B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395   E.wex 1613    e. wcel 1819    =/= wne 2652   _Vcvv 3109    \ cdif 3468    u. cun 3469    i^i cin 3470    C_ wss 3471   (/)c0 3793   class class class wbr 4456    X. cxp 5006  (class class class)co 6296    ^m cmap 7438    ~~ cen 7532    ~<_ cdom 7533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-er 7329  df-map 7440  df-en 7536  df-dom 7537
This theorem is referenced by:  mapdom3  7708  cfpwsdom  8976  hauspwdom  20127
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