Theorem wunco 10735

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 10730	. . . 4 ⊢ (𝜑 → dom 𝐵 ∈ 𝑈)
4		dmcoss 5967	. . . . 5 ⊢ dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵
5	4	a1i 11	. . . 4 ⊢ (𝜑 → dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵)
6	1, 3, 5	wunss 10714	. . 3 ⊢ (𝜑 → dom (𝐴 ∘ 𝐵) ∈ 𝑈)
7		wunop.2	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
8	1, 7	wunrn 10731	. . . 4 ⊢ (𝜑 → ran 𝐴 ∈ 𝑈)
9		rncoss 5969	. . . . 5 ⊢ ran (𝐴 ∘ 𝐵) ⊆ ran 𝐴
10	9	a1i 11	. . . 4 ⊢ (𝜑 → ran (𝐴 ∘ 𝐵) ⊆ ran 𝐴)
11	1, 8, 10	wunss 10714	. . 3 ⊢ (𝜑 → ran (𝐴 ∘ 𝐵) ∈ 𝑈)
12	1, 6, 11	wunxp 10726	. 2 ⊢ (𝜑 → (dom (𝐴 ∘ 𝐵) × ran (𝐴 ∘ 𝐵)) ∈ 𝑈)
13		relco 6112	. . 3 ⊢ Rel (𝐴 ∘ 𝐵)
14		relssdmrn 6273	. . 3 ⊢ (Rel (𝐴 ∘ 𝐵) → (𝐴 ∘ 𝐵) ⊆ (dom (𝐴 ∘ 𝐵) × ran (𝐴 ∘ 𝐵)))
15	13, 14	mp1i 14	. 2 ⊢ (𝜑 → (𝐴 ∘ 𝐵) ⊆ (dom (𝐴 ∘ 𝐵) × ran (𝐴 ∘ 𝐵)))
16	1, 12, 15	wunss 10714	1 ⊢ (𝜑 → (𝐴 ∘ 𝐵) ∈ 𝑈)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2146   ⊆ wss 3906   × cxp 5661  dom cdm 5663  ran crn 5664   ∘ ccom 5667  Rel wrel 5668  WUnicwun 10702

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

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 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-tr 5221  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-wun 10704

This theorem is used by: (None)