Theorem wuncnv 10486

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

Hypotheses

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

Assertion

Ref	Expression
wuncnv	⊢ (𝜑 → ◡𝐴 ∈ 𝑈)

Proof of Theorem wuncnv

Step	Hyp	Ref	Expression
1		wun0.1	. 2 ⊢ (𝜑 → 𝑈 ∈ WUni)
2		wunop.2	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
3	1, 2	wunrn 10485	. . 3 ⊢ (𝜑 → ran 𝐴 ∈ 𝑈)
4	1, 2	wundm 10484	. . 3 ⊢ (𝜑 → dom 𝐴 ∈ 𝑈)
5	1, 3, 4	wunxp 10480	. 2 ⊢ (𝜑 → (ran 𝐴 × dom 𝐴) ∈ 𝑈)
6		cnvssrndm 6174	. . 3 ⊢ ◡𝐴 ⊆ (ran 𝐴 × dom 𝐴)
7	6	a1i 11	. 2 ⊢ (𝜑 → ◡𝐴 ⊆ (ran 𝐴 × dom 𝐴))
8	1, 5, 7	wunss 10468	1 ⊢ (𝜑 → ◡𝐴 ∈ 𝑈)

Syntax hints:   → wi 4   ∈ wcel 2106   ⊆ wss 3887   × cxp 5587  ^◡ccnv 5588  dom cdm 5589  ran crn 5590  WUnicwun 10456

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-tr 5192  df-xp 5595  df-rel 5596  df-cnv 5597  df-dm 5599  df-rn 5600  df-wun 10458

This theorem is referenced by:  wuntpos  10490  catcoppccl  17832  catcoppcclOLD  17833