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Theorem mapdom1g 6563
Description: Order-preserving property of set exponentiation. (Contributed by Jim Kingdon, 15-Jul-2022.)
Assertion
Ref Expression
mapdom1g  |-  ( ( A  ~<_  B  /\  C  e.  V )  ->  ( A  ^m  C )  ~<_  ( B  ^m  C ) )

Proof of Theorem mapdom1g
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 reldom 6462 . . . . . 6  |-  Rel  ~<_
21brrelex2i 4482 . . . . 5  |-  ( A  ~<_  B  ->  B  e.  _V )
3 domeng 6469 . . . . 5  |-  ( B  e.  _V  ->  ( A  ~<_  B  <->  E. x
( A  ~~  x  /\  x  C_  B ) ) )
42, 3syl 14 . . . 4  |-  ( A  ~<_  B  ->  ( A  ~<_  B 
<->  E. x ( A 
~~  x  /\  x  C_  B ) ) )
54ibi 174 . . 3  |-  ( A  ~<_  B  ->  E. x
( A  ~~  x  /\  x  C_  B ) )
65adantr 270 . 2  |-  ( ( A  ~<_  B  /\  C  e.  V )  ->  E. x
( A  ~~  x  /\  x  C_  B ) )
7 simpl 107 . . . 4  |-  ( ( A  ~~  x  /\  x  C_  B )  ->  A  ~~  x )
8 enrefg 6481 . . . . 5  |-  ( C  e.  V  ->  C  ~~  C )
98adantl 271 . . . 4  |-  ( ( A  ~<_  B  /\  C  e.  V )  ->  C  ~~  C )
10 mapen 6562 . . . 4  |-  ( ( A  ~~  x  /\  C  ~~  C )  -> 
( A  ^m  C
)  ~~  ( x  ^m  C ) )
117, 9, 10syl2anr 284 . . 3  |-  ( ( ( A  ~<_  B  /\  C  e.  V )  /\  ( A  ~~  x  /\  x  C_  B ) )  ->  ( A  ^m  C )  ~~  (
x  ^m  C )
)
122ad2antrr 472 . . . . 5  |-  ( ( ( A  ~<_  B  /\  C  e.  V )  /\  ( A  ~~  x  /\  x  C_  B ) )  ->  B  e.  _V )
13 simprr 499 . . . . 5  |-  ( ( ( A  ~<_  B  /\  C  e.  V )  /\  ( A  ~~  x  /\  x  C_  B ) )  ->  x  C_  B
)
14 mapss 6448 . . . . 5  |-  ( ( B  e.  _V  /\  x  C_  B )  -> 
( x  ^m  C
)  C_  ( B  ^m  C ) )
1512, 13, 14syl2anc 403 . . . 4  |-  ( ( ( A  ~<_  B  /\  C  e.  V )  /\  ( A  ~~  x  /\  x  C_  B ) )  ->  ( x  ^m  C )  C_  ( B  ^m  C ) )
16 fnmap 6412 . . . . . . 7  |-  ^m  Fn  ( _V  X.  _V )
17 elex 2630 . . . . . . 7  |-  ( C  e.  V  ->  C  e.  _V )
18 fnovex 5682 . . . . . . 7  |-  ( (  ^m  Fn  ( _V 
X.  _V )  /\  B  e.  _V  /\  C  e. 
_V )  ->  ( B  ^m  C )  e. 
_V )
1916, 2, 17, 18mp3an3an 1279 . . . . . 6  |-  ( ( A  ~<_  B  /\  C  e.  V )  ->  ( B  ^m  C )  e. 
_V )
20 ssdomg 6495 . . . . . 6  |-  ( ( B  ^m  C )  e.  _V  ->  (
( x  ^m  C
)  C_  ( B  ^m  C )  ->  (
x  ^m  C )  ~<_  ( B  ^m  C ) ) )
2119, 20syl 14 . . . . 5  |-  ( ( A  ~<_  B  /\  C  e.  V )  ->  (
( x  ^m  C
)  C_  ( B  ^m  C )  ->  (
x  ^m  C )  ~<_  ( B  ^m  C ) ) )
2221adantr 270 . . . 4  |-  ( ( ( A  ~<_  B  /\  C  e.  V )  /\  ( A  ~~  x  /\  x  C_  B ) )  ->  ( (
x  ^m  C )  C_  ( B  ^m  C
)  ->  ( x  ^m  C )  ~<_  ( B  ^m  C ) ) )
2315, 22mpd 13 . . 3  |-  ( ( ( A  ~<_  B  /\  C  e.  V )  /\  ( A  ~~  x  /\  x  C_  B ) )  ->  ( x  ^m  C )  ~<_  ( B  ^m  C ) )
24 endomtr 6507 . . 3  |-  ( ( ( A  ^m  C
)  ~~  ( x  ^m  C )  /\  (
x  ^m  C )  ~<_  ( B  ^m  C ) )  ->  ( A  ^m  C )  ~<_  ( B  ^m  C ) )
2511, 23, 24syl2anc 403 . 2  |-  ( ( ( A  ~<_  B  /\  C  e.  V )  /\  ( A  ~~  x  /\  x  C_  B ) )  ->  ( A  ^m  C )  ~<_  ( B  ^m  C ) )
266, 25exlimddv 1826 1  |-  ( ( A  ~<_  B  /\  C  e.  V )  ->  ( A  ^m  C )  ~<_  ( B  ^m  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   E.wex 1426    e. wcel 1438   _Vcvv 2619    C_ wss 2999   class class class wbr 3845    X. cxp 4436    Fn wfn 5010  (class class class)co 5652    ^m cmap 6405    ~~ cen 6455    ~<_ cdom 6456
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-map 6407  df-en 6458  df-dom 6459
This theorem is referenced by: (None)
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