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Theorem mapsspw 7359
Description: Set exponentiation is a subset of the power set of the Cartesian product of its arguments. (Contributed by NM, 8-Dec-2006.) (Revised by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
mapsspw  |-  ( A  ^m  B )  C_  ~P ( B  X.  A
)

Proof of Theorem mapsspw
StepHypRef Expression
1 mapsspm 7357 . 2  |-  ( A  ^m  B )  C_  ( A  ^pm  B )
2 pmsspw 7358 . 2  |-  ( A 
^pm  B )  C_  ~P ( B  X.  A
)
31, 2sstri 3474 1  |-  ( A  ^m  B )  C_  ~P ( B  X.  A
)
Colors of variables: wff setvar class
Syntax hints:    C_ wss 3437   ~Pcpw 3969    X. cxp 4947  (class class class)co 6201    ^m cmap 7325    ^pm cpm 7326
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-fv 5535  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-1st 6688  df-2nd 6689  df-map 7327  df-pm 7328
This theorem is referenced by:  mapfi  7719  rankmapu  8197  grumap  9087  wrdexg  12363  wunfunc  14929
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