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Theorem mapss 7274
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 7253 . . . . . 6  |-  ( f  e.  ( A  ^m  C )  ->  f : C --> A )
21adantl 466 . . . . 5  |-  ( ( ( B  e.  V  /\  A  C_  B )  /\  f  e.  ( A  ^m  C ) )  ->  f : C
--> A )
3 simplr 754 . . . . 5  |-  ( ( ( B  e.  V  /\  A  C_  B )  /\  f  e.  ( A  ^m  C ) )  ->  A  C_  B
)
4 fss 5586 . . . . 5  |-  ( ( f : C --> A  /\  A  C_  B )  -> 
f : C --> B )
52, 3, 4syl2anc 661 . . . 4  |-  ( ( ( B  e.  V  /\  A  C_  B )  /\  f  e.  ( A  ^m  C ) )  ->  f : C
--> B )
6 simpll 753 . . . . 5  |-  ( ( ( B  e.  V  /\  A  C_  B )  /\  f  e.  ( A  ^m  C ) )  ->  B  e.  V )
7 elmapex 7252 . . . . . . 7  |-  ( f  e.  ( A  ^m  C )  ->  ( A  e.  _V  /\  C  e.  _V ) )
87simprd 463 . . . . . 6  |-  ( f  e.  ( A  ^m  C )  ->  C  e.  _V )
98adantl 466 . . . . 5  |-  ( ( ( B  e.  V  /\  A  C_  B )  /\  f  e.  ( A  ^m  C ) )  ->  C  e.  _V )
10 elmapg 7246 . . . . 5  |-  ( ( B  e.  V  /\  C  e.  _V )  ->  ( f  e.  ( B  ^m  C )  <-> 
f : C --> B ) )
116, 9, 10syl2anc 661 . . . 4  |-  ( ( ( B  e.  V  /\  A  C_  B )  /\  f  e.  ( A  ^m  C ) )  ->  ( f  e.  ( B  ^m  C
)  <->  f : C --> B ) )
125, 11mpbird 232 . . 3  |-  ( ( ( B  e.  V  /\  A  C_  B )  /\  f  e.  ( A  ^m  C ) )  ->  f  e.  ( B  ^m  C ) )
1312ex 434 . 2  |-  ( ( B  e.  V  /\  A  C_  B )  -> 
( f  e.  ( A  ^m  C )  ->  f  e.  ( B  ^m  C ) ) )
1413ssrdv 3381 1  |-  ( ( B  e.  V  /\  A  C_  B )  -> 
( A  ^m  C
)  C_  ( B  ^m  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1756   _Vcvv 2991    C_ wss 3347   -->wf 5433  (class class class)co 6110    ^m cmap 7233
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2739  df-rex 2740  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-op 3903  df-uni 4111  df-iun 4192  df-br 4312  df-opab 4370  df-mpt 4371  df-id 4655  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-fv 5445  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6596  df-2nd 6597  df-map 7235
This theorem is referenced by:  mapdom1  7495  ssfin3ds  8518  ingru  9001  resspsrbas  17506  resspsradd  17507  resspsrmul  17508  plyss  21686  eulerpartlem1  26769  eulerpartlemn  26783  diophrw  29120  diophin  29134  diophun  29135  eq0rabdioph  29138  eqrabdioph  29139  rabdiophlem1  29162  diophren  29175
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