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Theorem mapdom2 7740
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 463 . . . . . . . 8  |-  ( ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  /\  C  =  (/) )  ->  C  =  (/) )
21oveq1d 6303 . . . . . . 7  |-  ( ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  /\  C  =  (/) )  ->  ( C  ^m  A )  =  ( (/)  ^m  A ) )
3 simplr 761 . . . . . . . . . 10  |-  ( ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  /\  C  =  (/) )  ->  -.  ( A  =  (/)  /\  C  =  (/) ) )
4 idd 25 . . . . . . . . . . 11  |-  ( ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  /\  C  =  (/) )  ->  ( A  =  (/)  ->  A  =  (/) ) )
54, 1jctird 547 . . . . . . . . . 10  |-  ( ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  /\  C  =  (/) )  ->  ( A  =  (/)  ->  ( A  =  (/)  /\  C  =  (/) ) ) )
63, 5mtod 181 . . . . . . . . 9  |-  ( ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  /\  C  =  (/) )  ->  -.  A  =  (/) )
76neqned 2630 . . . . . . . 8  |-  ( ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  /\  C  =  (/) )  ->  A  =/=  (/) )
8 map0b 7507 . . . . . . . 8  |-  ( A  =/=  (/)  ->  ( (/)  ^m  A
)  =  (/) )
97, 8syl 17 . . . . . . 7  |-  ( ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  /\  C  =  (/) )  ->  ( (/) 
^m  A )  =  (/) )
102, 9eqtrd 2484 . . . . . 6  |-  ( ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  /\  C  =  (/) )  ->  ( C  ^m  A )  =  (/) )
11 ovex 6316 . . . . . . 7  |-  ( C  ^m  B )  e. 
_V
12110dom 7699 . . . . . 6  |-  (/)  ~<_  ( C  ^m  B )
1310, 12syl6eqbr 4439 . . . . 5  |-  ( ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  /\  C  =  (/) )  ->  ( C  ^m  A )  ~<_  ( C  ^m  B ) )
14 simpll 759 . . . . . . . 8  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  C  =/=  (/) )  ->  A  ~<_  B )
15 reldom 7572 . . . . . . . . . . 11  |-  Rel  ~<_
1615brrelex2i 4875 . . . . . . . . . 10  |-  ( A  ~<_  B  ->  B  e.  _V )
1716ad2antrr 731 . . . . . . . . 9  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  C  =/=  (/) )  ->  B  e.  _V )
18 domeng 7580 . . . . . . . . 9  |-  ( B  e.  _V  ->  ( A  ~<_  B  <->  E. x
( A  ~~  x  /\  x  C_  B ) ) )
1917, 18syl 17 . . . . . . . 8  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  C  =/=  (/) )  -> 
( A  ~<_  B  <->  E. x
( A  ~~  x  /\  x  C_  B ) ) )
2014, 19mpbid 214 . . . . . . 7  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  C  =/=  (/) )  ->  E. x ( A  ~~  x  /\  x  C_  B
) )
21 enrefg 7598 . . . . . . . . . . . 12  |-  ( C  e.  _V  ->  C  ~~  C )
2221ad2antlr 732 . . . . . . . . . . 11  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( C  =/=  (/)  /\  ( A  ~~  x  /\  x  C_  B ) ) )  ->  C  ~~  C
)
23 simprrl 773 . . . . . . . . . . 11  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( C  =/=  (/)  /\  ( A  ~~  x  /\  x  C_  B ) ) )  ->  A  ~~  x
)
24 mapen 7733 . . . . . . . . . . 11  |-  ( ( C  ~~  C  /\  A  ~~  x )  -> 
( C  ^m  A
)  ~~  ( C  ^m  x ) )
2522, 23, 24syl2anc 666 . . . . . . . . . 10  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( C  =/=  (/)  /\  ( A  ~~  x  /\  x  C_  B ) ) )  ->  ( C  ^m  A )  ~~  ( C  ^m  x ) )
26 ovex 6316 . . . . . . . . . . . . 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 6316 . . . . . . . . . . . . 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 763 . . . . . . . . . . . . 13  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( C  =/=  (/)  /\  ( A  ~~  x  /\  x  C_  B ) ) )  ->  C  =/=  (/) )
31 simplr 761 . . . . . . . . . . . . . 14  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( C  =/=  (/)  /\  ( A  ~~  x  /\  x  C_  B ) ) )  ->  C  e.  _V )
3216ad2antrr 731 . . . . . . . . . . . . . . 15  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( C  =/=  (/)  /\  ( A  ~~  x  /\  x  C_  B ) ) )  ->  B  e.  _V )
33 difexg 4550 . . . . . . . . . . . . . . 15  |-  ( B  e.  _V  ->  ( B  \  x )  e. 
_V )
3432, 33syl 17 . . . . . . . . . . . . . 14  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( C  =/=  (/)  /\  ( A  ~~  x  /\  x  C_  B ) ) )  ->  ( B  \  x )  e.  _V )
35 map0g 7508 . . . . . . . . . . . . . . . 16  |-  ( ( C  e.  _V  /\  ( B  \  x
)  e.  _V )  ->  ( ( C  ^m  ( B  \  x
) )  =  (/)  <->  ( C  =  (/)  /\  ( B  \  x )  =/=  (/) ) ) )
36 simpl 459 . . . . . . . . . . . . . . . 16  |-  ( ( C  =  (/)  /\  ( B  \  x )  =/=  (/) )  ->  C  =  (/) )
3735, 36syl6bi 232 . . . . . . . . . . . . . . 15  |-  ( ( C  e.  _V  /\  ( B  \  x
)  e.  _V )  ->  ( ( C  ^m  ( B  \  x
) )  =  (/)  ->  C  =  (/) ) )
3837necon3d 2644 . . . . . . . . . . . . . 14  |-  ( ( C  e.  _V  /\  ( B  \  x
)  e.  _V )  ->  ( C  =/=  (/)  ->  ( C  ^m  ( B  \  x ) )  =/=  (/) ) )
3931, 34, 38syl2anc 666 . . . . . . . . . . . . 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 7667 . . . . . . . . . . . 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 1267 . . . . . . . . . . 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 3047 . . . . . . . . . . . . . . 15  |-  x  e. 
_V
4443a1i 11 . . . . . . . . . . . . . 14  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( C  =/=  (/)  /\  ( A  ~~  x  /\  x  C_  B ) ) )  ->  x  e.  _V )
45 disjdif 3838 . . . . . . . . . . . . . . 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 7738 . . . . . . . . . . . . . 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 1270 . . . . . . . . . . . . 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 7617 . . . . . . . . . . . 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 774 . . . . . . . . . . . . . 14  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( C  =/=  (/)  /\  ( A  ~~  x  /\  x  C_  B ) ) )  ->  x  C_  B
)
51 undif 3847 . . . . . . . . . . . . . 14  |-  ( x 
C_  B  <->  ( x  u.  ( B  \  x
) )  =  B )
5250, 51sylib 200 . . . . . . . . . . . . 13  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( C  =/=  (/)  /\  ( A  ~~  x  /\  x  C_  B ) ) )  ->  ( x  u.  ( B  \  x
) )  =  B )
5352oveq2d 6304 . . . . . . . . . . . 12  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( C  =/=  (/)  /\  ( A  ~~  x  /\  x  C_  B ) ) )  ->  ( C  ^m  ( x  u.  ( B  \  x ) ) )  =  ( C  ^m  B ) )
5449, 53breqtrd 4426 . . . . . . . . . . 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 7625 . . . . . . . . . . 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 666 . . . . . . . . . 10  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( C  =/=  (/)  /\  ( A  ~~  x  /\  x  C_  B ) ) )  ->  ( C  ^m  x )  ~<_  ( C  ^m  B ) )
57 endomtr 7624 . . . . . . . . . 10  |-  ( ( ( C  ^m  A
)  ~~  ( C  ^m  x )  /\  ( C  ^m  x )  ~<_  ( C  ^m  B ) )  ->  ( C  ^m  A )  ~<_  ( C  ^m  B ) )
5825, 56, 57syl2anc 666 . . . . . . . . 9  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  ( C  =/=  (/)  /\  ( A  ~~  x  /\  x  C_  B ) ) )  ->  ( C  ^m  A )  ~<_  ( C  ^m  B ) )
5958expr 619 . . . . . . . 8  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  C  =/=  (/) )  -> 
( ( A  ~~  x  /\  x  C_  B
)  ->  ( C  ^m  A )  ~<_  ( C  ^m  B ) ) )
6059exlimdv 1778 . . . . . . 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 720 . . . . 5  |-  ( ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  /\  C  =/=  (/) )  ->  ( C  ^m  A )  ~<_  ( C  ^m  B ) )
6313, 62pm2.61dane 2710 . . . 4  |-  ( ( ( A  ~<_  B  /\  C  e.  _V )  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  ->  ( C  ^m  A )  ~<_  ( C  ^m  B ) )
6463an32s 812 . . 3  |-  ( ( ( A  ~<_  B  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  /\  C  e.  _V )  ->  ( C  ^m  A )  ~<_  ( C  ^m  B ) )
6564ex 436 . 2  |-  ( ( A  ~<_  B  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  -> 
( C  e.  _V  ->  ( C  ^m  A
)  ~<_  ( C  ^m  B ) ) )
66 reldmmap 7478 . . . 4  |-  Rel  dom  ^m
6766ovprc1 6319 . . 3  |-  ( -.  C  e.  _V  ->  ( C  ^m  A )  =  (/) )
6867, 12syl6eqbr 4439 . 2  |-  ( -.  C  e.  _V  ->  ( C  ^m  A )  ~<_  ( C  ^m  B
) )
6965, 68pm2.61d1 163 1  |-  ( ( A  ~<_  B  /\  -.  ( A  =  (/)  /\  C  =  (/) ) )  -> 
( C  ^m  A
)  ~<_  ( C  ^m  B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1443   E.wex 1662    e. wcel 1886    =/= wne 2621   _Vcvv 3044    \ cdif 3400    u. cun 3401    i^i cin 3402    C_ wss 3403   (/)c0 3730   class class class wbr 4401    X. cxp 4831  (class class class)co 6288    ^m cmap 7469    ~~ cen 7563    ~<_ cdom 7564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-1st 6790  df-2nd 6791  df-er 7360  df-map 7471  df-en 7567  df-dom 7568
This theorem is referenced by:  mapdom3  7741  cfpwsdom  9006  hauspwdom  20509
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