Theorem wunrn 10838

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 10815	. . 3 ⊢ (𝜑 → ∪ 𝐴 ∈ 𝑈)
4	1, 3	wununi 10815	. 2 ⊢ (𝜑 → ∪ ∪ 𝐴 ∈ 𝑈)
5		ssun2 4180	. . . 4 ⊢ ran 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴)
6		dmrnssfld 5962	. . . 4 ⊢ (dom 𝐴 ∪ ran 𝐴) ⊆ ∪ ∪ 𝐴
7	5, 6	sstri 3922	. . 3 ⊢ ran 𝐴 ⊆ ∪ ∪ 𝐴
8	7	a1i 11	. 2 ⊢ (𝜑 → ran 𝐴 ⊆ ∪ ∪ 𝐴)
9	1, 4, 8	wunss 10821	1 ⊢ (𝜑 → ran 𝐴 ∈ 𝑈)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147   ⊆ wss 3900   ∪ cun 4152  ∪ cuni 4870  dom cdm 5655  ran crn 5656  WUnicwun 10809

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-tr 5216  df-cnv 5663  df-dm 5665  df-rn 5666  df-wun 10811

This theorem is used by:  wuncnv  10839  wunfv  10841  wunco  10842  wuntpos  10843  wunstr  17392  wunfunc  18102  wunnat  18160  catcoppccl  18318  catcfuccl  18319  catcxpccl  18407