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Theorem mapsspm 7047
Description: Set exponentiation is a subset of partial maps. (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 27-Feb-2016.)
Assertion
Ref Expression
mapsspm  |-  ( A  ^m  B )  C_  ( A  ^pm  B )

Proof of Theorem mapsspm
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 elmapex 7037 . . . 4  |-  ( f  e.  ( A  ^m  B )  ->  ( A  e.  _V  /\  B  e.  _V ) )
21simprd 450 . . 3  |-  ( f  e.  ( A  ^m  B )  ->  B  e.  _V )
31simpld 446 . . 3  |-  ( f  e.  ( A  ^m  B )  ->  A  e.  _V )
4 elmapi 7038 . . 3  |-  ( f  e.  ( A  ^m  B )  ->  f : B --> A )
5 fpmg 7039 . . 3  |-  ( ( B  e.  _V  /\  A  e.  _V  /\  f : B --> A )  -> 
f  e.  ( A 
^pm  B ) )
62, 3, 4, 5syl3anc 1184 . 2  |-  ( f  e.  ( A  ^m  B )  ->  f  e.  ( A  ^pm  B
) )
76ssriv 3352 1  |-  ( A  ^m  B )  C_  ( A  ^pm  B )
Colors of variables: wff set class
Syntax hints:    e. wcel 1725   _Vcvv 2956    C_ wss 3320   -->wf 5450  (class class class)co 6081    ^m cmap 7018    ^pm cpm 7019
This theorem is referenced by:  mapsspw  7049  wunmap  8601  dvntaylp  20287  taylthlem1  20289  taylthlem2  20290
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-map 7020  df-pm 7021
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