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Theorem mapsspw 7078
Description: Set exponentiation is a subset of the power set of the cross 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 7076 . 2  |-  ( A  ^m  B )  C_  ( A  ^pm  B )
2 pmsspw 7077 . 2  |-  ( A 
^pm  B )  C_  ~P ( B  X.  A
)
31, 2sstri 3343 1  |-  ( A  ^m  B )  C_  ~P ( B  X.  A
)
Colors of variables: wff set class
Syntax hints:    C_ wss 3306   ~Pcpw 3823    X. cxp 4905  (class class class)co 6110    ^m cmap 7047    ^pm cpm 7048
This theorem is referenced by:  mapfi  7432  rankmapu  7833  grumap  8714  wrdexg  11770  wunfunc  14127
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-map 7049  df-pm 7050
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