Theorem wunrn 9873

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 9850	. . 3 ⊢ (𝜑 → ∪ 𝐴 ∈ 𝑈)
4	1, 3	wununi 9850	. 2 ⊢ (𝜑 → ∪ ∪ 𝐴 ∈ 𝑈)
5		ssun2 4006	. . . 4 ⊢ ran 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴)
6		dmrnssfld 5621	. . . 4 ⊢ (dom 𝐴 ∪ ran 𝐴) ⊆ ∪ ∪ 𝐴
7	5, 6	sstri 3836	. . 3 ⊢ ran 𝐴 ⊆ ∪ ∪ 𝐴
8	7	a1i 11	. 2 ⊢ (𝜑 → ran 𝐴 ⊆ ∪ ∪ 𝐴)
9	1, 4, 8	wunss 9856	1 ⊢ (𝜑 → ran 𝐴 ∈ 𝑈)

Syntax hints:   → wi 4   ∈ wcel 2164   ∪ cun 3796   ⊆ wss 3798  ∪ cuni 4660  dom cdm 5346  ran crn 5347  WUnicwun 9844

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pr 5129

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-tr 4978  df-cnv 5354  df-dm 5356  df-rn 5357  df-wun 9846

This theorem is referenced by:  wuncnv  9874  wunfv  9876  wunco  9877  wuntpos  9878  wunstr  16253  wunfunc  16918  wunnat  16975  catcoppccl  17117  catcfuccl  17118  catcxpccl  17207