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Theorem grumap 8675
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 957 . 2  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  U  e.  Univ )
2 gruxp 8674 . . . 4  |-  ( ( U  e.  Univ  /\  B  e.  U  /\  A  e.  U )  ->  ( B  X.  A )  e.  U )
323com23 1159 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  ( B  X.  A )  e.  U )
4 grupw 8662 . . 3  |-  ( ( U  e.  Univ  /\  ( B  X.  A )  e.  U )  ->  ~P ( B  X.  A
)  e.  U )
51, 3, 4syl2anc 643 . 2  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  ~P ( B  X.  A
)  e.  U )
6 mapsspw 7041 . . 3  |-  ( A  ^m  B )  C_  ~P ( B  X.  A
)
76a1i 11 . 2  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  ( A  ^m  B )  C_  ~P ( B  X.  A
) )
8 gruss 8663 . 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 1184 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 936    e. wcel 1725    C_ wss 3312   ~Pcpw 3791    X. cxp 4868  (class class class)co 6073    ^m cmap 7010   Univcgru 8657
This theorem is referenced by:  gruixp  8676
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-map 7012  df-pm 7013  df-gru 8658
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