Theorem wunrn 10630

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 10607	. . 3 ⊢ (𝜑 → ∪ 𝐴 ∈ 𝑈)
4	1, 3	wununi 10607	. 2 ⊢ (𝜑 → ∪ ∪ 𝐴 ∈ 𝑈)
5		ssun2 4130	. . . 4 ⊢ ran 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴)
6		dmrnssfld 5920	. . . 4 ⊢ (dom 𝐴 ∪ ran 𝐴) ⊆ ∪ ∪ 𝐴
7	5, 6	sstri 3941	. . 3 ⊢ ran 𝐴 ⊆ ∪ ∪ 𝐴
8	7	a1i 11	. 2 ⊢ (𝜑 → ran 𝐴 ⊆ ∪ ∪ 𝐴)
9	1, 4, 8	wunss 10613	1 ⊢ (𝜑 → ran 𝐴 ∈ 𝑈)

Syntax hints:   → wi 4   ∈ wcel 2113   ∪ cun 3897   ⊆ wss 3899  ∪ cuni 4860  dom cdm 5621  ran crn 5622  WUnicwun 10601

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-tr 5203  df-cnv 5629  df-dm 5631  df-rn 5632  df-wun 10603

This theorem is referenced by:  wuncnv  10631  wunfv  10633  wunco  10634  wuntpos  10635  wunstr  17109  wunfunc  17818  wunnat  17876  catcoppccl  18034  catcfuccl  18035  catcxpccl  18123