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Theorem mapss 6592
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 ) )

Proof of Theorem mapss
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 elmapi 6571 . . . . . 6  |-  ( f  e.  ( A  ^m  C )  ->  f : C --> A )
21adantl 275 . . . . 5  |-  ( ( ( B  e.  V  /\  A  C_  B )  /\  f  e.  ( A  ^m  C ) )  ->  f : C
--> A )
3 simplr 520 . . . . 5  |-  ( ( ( B  e.  V  /\  A  C_  B )  /\  f  e.  ( A  ^m  C ) )  ->  A  C_  B
)
42, 3fssd 5292 . . . 4  |-  ( ( ( B  e.  V  /\  A  C_  B )  /\  f  e.  ( A  ^m  C ) )  ->  f : C
--> B )
5 simpll 519 . . . . 5  |-  ( ( ( B  e.  V  /\  A  C_  B )  /\  f  e.  ( A  ^m  C ) )  ->  B  e.  V )
6 elmapex 6570 . . . . . . 7  |-  ( f  e.  ( A  ^m  C )  ->  ( A  e.  _V  /\  C  e.  _V ) )
76simprd 113 . . . . . 6  |-  ( f  e.  ( A  ^m  C )  ->  C  e.  _V )
87adantl 275 . . . . 5  |-  ( ( ( B  e.  V  /\  A  C_  B )  /\  f  e.  ( A  ^m  C ) )  ->  C  e.  _V )
95, 8elmapd 6563 . . . 4  |-  ( ( ( B  e.  V  /\  A  C_  B )  /\  f  e.  ( A  ^m  C ) )  ->  ( f  e.  ( B  ^m  C
)  <->  f : C --> B ) )
104, 9mpbird 166 . . 3  |-  ( ( ( B  e.  V  /\  A  C_  B )  /\  f  e.  ( A  ^m  C ) )  ->  f  e.  ( B  ^m  C ) )
1110ex 114 . 2  |-  ( ( B  e.  V  /\  A  C_  B )  -> 
( f  e.  ( A  ^m  C )  ->  f  e.  ( B  ^m  C ) ) )
1211ssrdv 3107 1  |-  ( ( B  e.  V  /\  A  C_  B )  -> 
( A  ^m  C
)  C_  ( B  ^m  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    e. wcel 1481   _Vcvv 2689    C_ wss 3075   -->wf 5126  (class class class)co 5781    ^m cmap 6549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2913  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-opab 3997  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-fv 5138  df-ov 5784  df-oprab 5785  df-mpo 5786  df-map 6551
This theorem is referenced by:  mapdom1g  6748
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