Theorem wunrn 10706

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 10683	. . 3 ⊢ (𝜑 → ∪ 𝐴 ∈ 𝑈)
4	1, 3	wununi 10683	. 2 ⊢ (𝜑 → ∪ ∪ 𝐴 ∈ 𝑈)
5		ssun2 4169	. . . 4 ⊢ ran 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴)
6		dmrnssfld 5961	. . . 4 ⊢ (dom 𝐴 ∪ ran 𝐴) ⊆ ∪ ∪ 𝐴
7	5, 6	sstri 3987	. . 3 ⊢ ran 𝐴 ⊆ ∪ ∪ 𝐴
8	7	a1i 11	. 2 ⊢ (𝜑 → ran 𝐴 ⊆ ∪ ∪ 𝐴)
9	1, 4, 8	wunss 10689	1 ⊢ (𝜑 → ran 𝐴 ∈ 𝑈)

Syntax hints:   → wi 4   ∈ wcel 2106   ∪ cun 3942   ⊆ wss 3944  ∪ cuni 4901  dom cdm 5669  ran crn 5670  WUnicwun 10677

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-tr 5259  df-cnv 5677  df-dm 5679  df-rn 5680  df-wun 10679

This theorem is referenced by:  wuncnv  10707  wunfv  10709  wunco  10710  wuntpos  10711  wunstr  17103  wunfunc  17831  wunfuncOLD  17832  wunnat  17889  wunnatOLD  17890  catcoppccl  18049  catcoppcclOLD  18050  catcfuccl  18051  catcfucclOLD  18052  catcxpccl  18141  catcxpcclOLD  18142