Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  wunfv Structured version   Visualization version   GIF version

Theorem wunfv 10143
 Description: A weak universe is closed under the function value operator. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
wun0.1 (𝜑𝑈 ∈ WUni)
wunop.2 (𝜑𝐴𝑈)
Assertion
Ref Expression
wunfv (𝜑 → (𝐴𝐵) ∈ 𝑈)

Proof of Theorem wunfv
StepHypRef Expression
1 wun0.1 . 2 (𝜑𝑈 ∈ WUni)
2 wunop.2 . . . 4 (𝜑𝐴𝑈)
31, 2wunrn 10140 . . 3 (𝜑 → ran 𝐴𝑈)
41, 3wununi 10117 . 2 (𝜑 ran 𝐴𝑈)
5 fvssunirn 6681 . . 3 (𝐴𝐵) ⊆ ran 𝐴
65a1i 11 . 2 (𝜑 → (𝐴𝐵) ⊆ ran 𝐴)
71, 4, 6wunss 10123 1 (𝜑 → (𝐴𝐵) ∈ 𝑈)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2114   ⊆ wss 3908  ∪ cuni 4813  ran crn 5533  ‘cfv 6334  WUnicwun 10111 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-tr 5149  df-cnv 5540  df-dm 5542  df-rn 5543  df-iota 6293  df-fv 6342  df-wun 10113 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator