Theorem wuncnv 10749

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 10748	. . 3 ⊢ (𝜑 → ran 𝐴 ∈ 𝑈)
4	1, 2	wundm 10747	. . 3 ⊢ (𝜑 → dom 𝐴 ∈ 𝑈)
5	1, 3, 4	wunxp 10743	. 2 ⊢ (𝜑 → (ran 𝐴 × dom 𝐴) ∈ 𝑈)
6		cnvssrndm 6265	. . 3 ⊢ ◡𝐴 ⊆ (ran 𝐴 × dom 𝐴)
7	6	a1i 11	. 2 ⊢ (𝜑 → ◡𝐴 ⊆ (ran 𝐴 × dom 𝐴))
8	1, 5, 7	wunss 10731	1 ⊢ (𝜑 → ◡𝐴 ∈ 𝑈)

Syntax hints:   → wi 4   ∈ wcel 2109   ⊆ wss 3931   × cxp 5657  ^◡ccnv 5658  dom cdm 5659  ran crn 5660  WUnicwun 10719

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-tr 5235  df-xp 5665  df-rel 5666  df-cnv 5667  df-dm 5669  df-rn 5670  df-wun 10721

This theorem is referenced by:  wuntpos  10753  catcoppccl  18135