ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  mapdom1g Unicode version

Theorem mapdom1g 6860
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 6758 . . . . . 6  |-  Rel  ~<_
21brrelex2i 4682 . . . . 5  |-  ( A  ~<_  B  ->  B  e.  _V )
3 domeng 6765 . . . . 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 176 . . 3  |-  ( A  ~<_  B  ->  E. x
( A  ~~  x  /\  x  C_  B ) )
65adantr 276 . 2  |-  ( ( A  ~<_  B  /\  C  e.  V )  ->  E. x
( A  ~~  x  /\  x  C_  B ) )
7 simpl 109 . . . 4  |-  ( ( A  ~~  x  /\  x  C_  B )  ->  A  ~~  x )
8 enrefg 6777 . . . . 5  |-  ( C  e.  V  ->  C  ~~  C )
98adantl 277 . . . 4  |-  ( ( A  ~<_  B  /\  C  e.  V )  ->  C  ~~  C )
10 mapen 6859 . . . 4  |-  ( ( A  ~~  x  /\  C  ~~  C )  -> 
( A  ^m  C
)  ~~  ( x  ^m  C ) )
117, 9, 10syl2anr 290 . . 3  |-  ( ( ( A  ~<_  B  /\  C  e.  V )  /\  ( A  ~~  x  /\  x  C_  B ) )  ->  ( A  ^m  C )  ~~  (
x  ^m  C )
)
122ad2antrr 488 . . . . 5  |-  ( ( ( A  ~<_  B  /\  C  e.  V )  /\  ( A  ~~  x  /\  x  C_  B ) )  ->  B  e.  _V )
13 simprr 531 . . . . 5  |-  ( ( ( A  ~<_  B  /\  C  e.  V )  /\  ( A  ~~  x  /\  x  C_  B ) )  ->  x  C_  B
)
14 mapss 6704 . . . . 5  |-  ( ( B  e.  _V  /\  x  C_  B )  -> 
( x  ^m  C
)  C_  ( B  ^m  C ) )
1512, 13, 14syl2anc 411 . . . 4  |-  ( ( ( A  ~<_  B  /\  C  e.  V )  /\  ( A  ~~  x  /\  x  C_  B ) )  ->  ( x  ^m  C )  C_  ( B  ^m  C ) )
16 fnmap 6668 . . . . . . 7  |-  ^m  Fn  ( _V  X.  _V )
17 elex 2760 . . . . . . 7  |-  ( C  e.  V  ->  C  e.  _V )
18 fnovex 5921 . . . . . . 7  |-  ( (  ^m  Fn  ( _V 
X.  _V )  /\  B  e.  _V  /\  C  e. 
_V )  ->  ( B  ^m  C )  e. 
_V )
1916, 2, 17, 18mp3an3an 1353 . . . . . 6  |-  ( ( A  ~<_  B  /\  C  e.  V )  ->  ( B  ^m  C )  e. 
_V )
20 ssdomg 6791 . . . . . 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 276 . . . 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 6803 . . 3  |-  ( ( ( A  ^m  C
)  ~~  ( x  ^m  C )  /\  (
x  ^m  C )  ~<_  ( B  ^m  C ) )  ->  ( A  ^m  C )  ~<_  ( B  ^m  C ) )
2511, 23, 24syl2anc 411 . 2  |-  ( ( ( A  ~<_  B  /\  C  e.  V )  /\  ( A  ~~  x  /\  x  C_  B ) )  ->  ( A  ^m  C )  ~<_  ( B  ^m  C ) )
266, 25exlimddv 1908 1  |-  ( ( A  ~<_  B  /\  C  e.  V )  ->  ( A  ^m  C )  ~<_  ( B  ^m  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105   E.wex 1502    e. wcel 2158   _Vcvv 2749    C_ wss 3141   class class class wbr 4015    X. cxp 4636    Fn wfn 5223  (class class class)co 5888    ^m cmap 6661    ~~ cen 6751    ~<_ cdom 6752
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-map 6663  df-en 6754  df-dom 6755
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator