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Theorem grumap 8430
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 8429 . . . 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 8417 . . 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 6803 . . 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 8418 . 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 1684    C_ wss 3152   ~Pcpw 3625    X. cxp 4687  (class class class)co 5858    ^m cmap 6772   Univcgru 8412
This theorem is referenced by:  gruixp  8431  prismorcsetlemb  25913
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-map 6774  df-pm 6775  df-gru 8413
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