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Theorem wunrn 9502
Description: A weak universe is closed under the range operator. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
wun0.1 (𝜑𝑈 ∈ WUni)
wunop.2 (𝜑𝐴𝑈)
Assertion
Ref Expression
wunrn (𝜑 → ran 𝐴𝑈)

Proof of Theorem wunrn
StepHypRef Expression
1 wun0.1 . 2 (𝜑𝑈 ∈ WUni)
2 wunop.2 . . . 4 (𝜑𝐴𝑈)
31, 2wununi 9479 . . 3 (𝜑 𝐴𝑈)
41, 3wununi 9479 . 2 (𝜑 𝐴𝑈)
5 ssun2 3760 . . . 4 ran 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴)
6 dmrnssfld 5349 . . . 4 (dom 𝐴 ∪ ran 𝐴) ⊆ 𝐴
75, 6sstri 3596 . . 3 ran 𝐴 𝐴
87a1i 11 . 2 (𝜑 → ran 𝐴 𝐴)
91, 4, 8wunss 9485 1 (𝜑 → ran 𝐴𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1987  cun 3557  wss 3559   cuni 4407  dom cdm 5079  ran crn 5080  WUnicwun 9473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-tr 4718  df-cnv 5087  df-dm 5089  df-rn 5090  df-wun 9475
This theorem is referenced by:  wuncnv  9503  wunfv  9505  wunco  9506  wuntpos  9507  wunstr  15810  wunfunc  16487  wunnat  16544  catcoppccl  16686  catcfuccl  16687  catcxpccl  16775
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