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Theorem grumap 8446
Description: A Grothendieck's universe contains all powers of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
Assertion
Ref Expression
grumap  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  ( A  ^m  B )  e.  U )

Proof of Theorem grumap
StepHypRef Expression
1 simp1 955 . 2  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  U  e.  Univ )
2 gruxp 8445 . . . 4  |-  ( ( U  e.  Univ  /\  B  e.  U  /\  A  e.  U )  ->  ( B  X.  A )  e.  U )
323com23 1157 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  ( B  X.  A )  e.  U )
4 grupw 8433 . . 3  |-  ( ( U  e.  Univ  /\  ( B  X.  A )  e.  U )  ->  ~P ( B  X.  A
)  e.  U )
51, 3, 4syl2anc 642 . 2  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  ~P ( B  X.  A
)  e.  U )
6 mapsspw 6819 . . 3  |-  ( A  ^m  B )  C_  ~P ( B  X.  A
)
76a1i 10 . 2  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  ( A  ^m  B )  C_  ~P ( B  X.  A
) )
8 gruss 8434 . 2  |-  ( ( U  e.  Univ  /\  ~P ( B  X.  A
)  e.  U  /\  ( A  ^m  B ) 
C_  ~P ( B  X.  A ) )  -> 
( A  ^m  B
)  e.  U )
91, 5, 7, 8syl3anc 1182 1  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  ( A  ^m  B )  e.  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    e. wcel 1696    C_ wss 3165   ~Pcpw 3638    X. cxp 4703  (class class class)co 5874    ^m cmap 6788   Univcgru 8428
This theorem is referenced by:  gruixp  8447  prismorcsetlemb  26016
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-map 6790  df-pm 6791  df-gru 8429
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