Theorem wunco 10771

Description: A weak universe is closed under composition. (Contributed by Mario Carneiro, 12-Jan-2017.)

Hypotheses

Ref	Expression
wun0.1	⊢ (𝜑 → 𝑈 ∈ WUni)
wunop.2	⊢ (𝜑 → 𝐴 ∈ 𝑈)
wunco.3	⊢ (𝜑 → 𝐵 ∈ 𝑈)

Assertion

Ref	Expression
wunco	⊢ (𝜑 → (𝐴 ∘ 𝐵) ∈ 𝑈)

Proof of Theorem wunco

Step	Hyp	Ref	Expression
1		wun0.1	. 2 ⊢ (𝜑 → 𝑈 ∈ WUni)
2		wunco.3	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑈)
3	1, 2	wundm 10766	. . . 4 ⊢ (𝜑 → dom 𝐵 ∈ 𝑈)
4		dmcoss 5988	. . . . 5 ⊢ dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵
5	4	a1i 11	. . . 4 ⊢ (𝜑 → dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵)
6	1, 3, 5	wunss 10750	. . 3 ⊢ (𝜑 → dom (𝐴 ∘ 𝐵) ∈ 𝑈)
7		wunop.2	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
8	1, 7	wunrn 10767	. . . 4 ⊢ (𝜑 → ran 𝐴 ∈ 𝑈)
9		rncoss 5989	. . . . 5 ⊢ ran (𝐴 ∘ 𝐵) ⊆ ran 𝐴
10	9	a1i 11	. . . 4 ⊢ (𝜑 → ran (𝐴 ∘ 𝐵) ⊆ ran 𝐴)
11	1, 8, 10	wunss 10750	. . 3 ⊢ (𝜑 → ran (𝐴 ∘ 𝐵) ∈ 𝑈)
12	1, 6, 11	wunxp 10762	. 2 ⊢ (𝜑 → (dom (𝐴 ∘ 𝐵) × ran (𝐴 ∘ 𝐵)) ∈ 𝑈)
13		relco 6129	. . 3 ⊢ Rel (𝐴 ∘ 𝐵)
14		relssdmrn 6290	. . 3 ⊢ (Rel (𝐴 ∘ 𝐵) → (𝐴 ∘ 𝐵) ⊆ (dom (𝐴 ∘ 𝐵) × ran (𝐴 ∘ 𝐵)))
15	13, 14	mp1i 13	. 2 ⊢ (𝜑 → (𝐴 ∘ 𝐵) ⊆ (dom (𝐴 ∘ 𝐵) × ran (𝐴 ∘ 𝐵)))
16	1, 12, 15	wunss 10750	1 ⊢ (𝜑 → (𝐴 ∘ 𝐵) ∈ 𝑈)

Syntax hints:   → wi 4   ∈ wcel 2106   ⊆ wss 3963   × cxp 5687  dom cdm 5689  ran crn 5690   ∘ ccom 5693  Rel wrel 5694  WUnicwun 10738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-wun 10740

This theorem is referenced by: (None)