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Theorem mapsspw 7016
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 7014 . 2  |-  ( A  ^m  B )  C_  ( A  ^pm  B )
2 pmsspw 7015 . 2  |-  ( A 
^pm  B )  C_  ~P ( B  X.  A
)
31, 2sstri 3325 1  |-  ( A  ^m  B )  C_  ~P ( B  X.  A
)
Colors of variables: wff set class
Syntax hints:    C_ wss 3288   ~Pcpw 3767    X. cxp 4843  (class class class)co 6048    ^m cmap 6985    ^pm cpm 6986
This theorem is referenced by:  mapfi  7369  grumap  8647  wrdexg  11702  wunfunc  14059
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-map 6987  df-pm 6988
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