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Theorem wunmap 9015
Description: A weak universe is closed under mappings. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
wun0.1  |-  ( ph  ->  U  e. WUni )
wunop.2  |-  ( ph  ->  A  e.  U )
wunop.3  |-  ( ph  ->  B  e.  U )
Assertion
Ref Expression
wunmap  |-  ( ph  ->  ( A  ^m  B
)  e.  U )

Proof of Theorem wunmap
StepHypRef Expression
1 wun0.1 . 2  |-  ( ph  ->  U  e. WUni )
2 wunop.2 . . 3  |-  ( ph  ->  A  e.  U )
3 wunop.3 . . 3  |-  ( ph  ->  B  e.  U )
41, 2, 3wunpm 9014 . 2  |-  ( ph  ->  ( A  ^pm  B
)  e.  U )
5 mapsspm 7371 . . 3  |-  ( A  ^m  B )  C_  ( A  ^pm  B )
65a1i 11 . 2  |-  ( ph  ->  ( A  ^m  B
)  C_  ( A  ^pm  B ) )
71, 4, 6wunss 9001 1  |-  ( ph  ->  ( A  ^m  B
)  e.  U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1826    C_ wss 3389  (class class class)co 6196    ^m cmap 7338    ^pm cpm 7339  WUnicwun 8989
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-1st 6699  df-2nd 6700  df-map 7340  df-pm 7341  df-wun 8991
This theorem is referenced by:  wunf  9016  tskmap  9077  wunfunc  15305  wunnat  15362  catcfuccl  15505
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