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Theorem mapss 6806
Description: Subset inheritance for set exponentiation. Theorem 99 of [Suppes] p. 89. (Contributed by NM, 10-Dec-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
mapss  |-  ( ( B  e.  V  /\  A  C_  B )  -> 
( A  ^m  C
)  C_  ( B  ^m  C ) )
Dummy variable  f is distinct from all other variables.

Proof of Theorem mapss
StepHypRef Expression
1 elmapi 6788 . . . . . 6  |-  ( f  e.  ( A  ^m  C )  ->  f : C --> A )
21adantl 454 . . . . 5  |-  ( ( ( B  e.  V  /\  A  C_  B )  /\  f  e.  ( A  ^m  C ) )  ->  f : C
--> A )
3 simplr 733 . . . . 5  |-  ( ( ( B  e.  V  /\  A  C_  B )  /\  f  e.  ( A  ^m  C ) )  ->  A  C_  B
)
4 fss 5363 . . . . 5  |-  ( ( f : C --> A  /\  A  C_  B )  -> 
f : C --> B )
52, 3, 4syl2anc 644 . . . 4  |-  ( ( ( B  e.  V  /\  A  C_  B )  /\  f  e.  ( A  ^m  C ) )  ->  f : C
--> B )
6 simpll 732 . . . . 5  |-  ( ( ( B  e.  V  /\  A  C_  B )  /\  f  e.  ( A  ^m  C ) )  ->  B  e.  V )
7 elmapex 6787 . . . . . . 7  |-  ( f  e.  ( A  ^m  C )  ->  ( A  e.  _V  /\  C  e.  _V ) )
87simprd 451 . . . . . 6  |-  ( f  e.  ( A  ^m  C )  ->  C  e.  _V )
98adantl 454 . . . . 5  |-  ( ( ( B  e.  V  /\  A  C_  B )  /\  f  e.  ( A  ^m  C ) )  ->  C  e.  _V )
10 elmapg 6781 . . . . 5  |-  ( ( B  e.  V  /\  C  e.  _V )  ->  ( f  e.  ( B  ^m  C )  <-> 
f : C --> B ) )
116, 9, 10syl2anc 644 . . . 4  |-  ( ( ( B  e.  V  /\  A  C_  B )  /\  f  e.  ( A  ^m  C ) )  ->  ( f  e.  ( B  ^m  C
)  <->  f : C --> B ) )
125, 11mpbird 225 . . 3  |-  ( ( ( B  e.  V  /\  A  C_  B )  /\  f  e.  ( A  ^m  C ) )  ->  f  e.  ( B  ^m  C ) )
1312ex 425 . 2  |-  ( ( B  e.  V  /\  A  C_  B )  -> 
( f  e.  ( A  ^m  C )  ->  f  e.  ( B  ^m  C ) ) )
1413ssrdv 3187 1  |-  ( ( B  e.  V  /\  A  C_  B )  -> 
( A  ^m  C
)  C_  ( B  ^m  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    e. wcel 1685   _Vcvv 2790    C_ wss 3154   -->wf 5218  (class class class)co 5820    ^m cmap 6768
This theorem is referenced by:  mapdom1  7022  ssfin3ds  7952  ingru  8433  resspsrbas  16154  resspsradd  16155  resspsrmul  16156  plyss  19576  negveudr  25069  subclrvd  25074  diophrw  26238  diophin  26252  diophun  26253  eq0rabdioph  26256  eqrabdioph  26257  rabdiophlem1  26282  diophren  26296
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5224  df-fn 5225  df-f 5226  df-fv 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-map 6770
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