Theorem wunrn 10308

Description: A weak universe is closed under the range operator. (Contributed by Mario Carneiro, 2-Jan-2017.)

Hypotheses

Ref	Expression
wun0.1	⊢ (𝜑 → 𝑈 ∈ WUni)
wunop.2	⊢ (𝜑 → 𝐴 ∈ 𝑈)

Assertion

Ref	Expression
wunrn	⊢ (𝜑 → ran 𝐴 ∈ 𝑈)

Proof of Theorem wunrn

Step	Hyp	Ref	Expression
1		wun0.1	. 2 ⊢ (𝜑 → 𝑈 ∈ WUni)
2		wunop.2	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
3	1, 2	wununi 10285	. . 3 ⊢ (𝜑 → ∪ 𝐴 ∈ 𝑈)
4	1, 3	wununi 10285	. 2 ⊢ (𝜑 → ∪ ∪ 𝐴 ∈ 𝑈)
5		ssun2 4073	. . . 4 ⊢ ran 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴)
6		dmrnssfld 5824	. . . 4 ⊢ (dom 𝐴 ∪ ran 𝐴) ⊆ ∪ ∪ 𝐴
7	5, 6	sstri 3896	. . 3 ⊢ ran 𝐴 ⊆ ∪ ∪ 𝐴
8	7	a1i 11	. 2 ⊢ (𝜑 → ran 𝐴 ⊆ ∪ ∪ 𝐴)
9	1, 4, 8	wunss 10291	1 ⊢ (𝜑 → ran 𝐴 ∈ 𝑈)

Syntax hints:   → wi 4   ∈ wcel 2112   ∪ cun 3851   ⊆ wss 3853  ∪ cuni 4805  dom cdm 5536  ran crn 5537  WUnicwun 10279

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-11 2160  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pr 5307

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-ne 2933  df-ral 3056  df-rab 3060  df-v 3400  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-br 5040  df-opab 5102  df-tr 5147  df-cnv 5544  df-dm 5546  df-rn 5547  df-wun 10281

This theorem is referenced by:  wuncnv  10309  wunfv  10311  wunco  10312  wuntpos  10313  wunstr  16689  wunfunc  17359  wunfuncOLD  17360  wunnat  17417  wunnatOLD  17418  catcoppccl  17577  catcoppcclOLD  17578  catcfuccl  17579  catcfucclOLD  17580  catcxpccl  17668  catcxpcclOLD  17669