MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  wunmap Structured version   Unicode version

Theorem wunmap 9007
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 9006 . 2  |-  ( ph  ->  ( A  ^pm  B
)  e.  U )
5 mapsspm 7359 . . 3  |-  ( A  ^m  B )  C_  ( A  ^pm  B )
65a1i 11 . 2  |-  ( ph  ->  ( A  ^m  B
)  C_  ( A  ^pm  B ) )
71, 4, 6wunss 8993 1  |-  ( ph  ->  ( A  ^m  B
)  e.  U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1758    C_ wss 3439  (class class class)co 6203    ^m cmap 7327    ^pm cpm 7328  WUnicwun 8981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-1st 6690  df-2nd 6691  df-map 7329  df-pm 7330  df-wun 8983
This theorem is referenced by:  wunf  9008  tskmap  9069  wunfunc  14931  wunnat  14988  catcfuccl  15099
  Copyright terms: Public domain W3C validator