Theorem wunco 10144

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 10139	. . . 4 ⊢ (𝜑 → dom 𝐵 ∈ 𝑈)
4		dmcoss 5807	. . . . 5 ⊢ dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵
5	4	a1i 11	. . . 4 ⊢ (𝜑 → dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵)
6	1, 3, 5	wunss 10123	. . 3 ⊢ (𝜑 → dom (𝐴 ∘ 𝐵) ∈ 𝑈)
7		wunop.2	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
8	1, 7	wunrn 10140	. . . 4 ⊢ (𝜑 → ran 𝐴 ∈ 𝑈)
9		rncoss 5808	. . . . 5 ⊢ ran (𝐴 ∘ 𝐵) ⊆ ran 𝐴
10	9	a1i 11	. . . 4 ⊢ (𝜑 → ran (𝐴 ∘ 𝐵) ⊆ ran 𝐴)
11	1, 8, 10	wunss 10123	. . 3 ⊢ (𝜑 → ran (𝐴 ∘ 𝐵) ∈ 𝑈)
12	1, 6, 11	wunxp 10135	. 2 ⊢ (𝜑 → (dom (𝐴 ∘ 𝐵) × ran (𝐴 ∘ 𝐵)) ∈ 𝑈)
13		relco 6064	. . 3 ⊢ Rel (𝐴 ∘ 𝐵)
14		relssdmrn 6088	. . 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 10123	1 ⊢ (𝜑 → (𝐴 ∘ 𝐵) ∈ 𝑈)

Syntax hints:   → wi 4   ∈ wcel 2111   ⊆ wss 3881   × cxp 5517  dom cdm 5519  ran crn 5520   ∘ ccom 5523  Rel wrel 5524  WUnicwun 10111

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-tr 5137  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-wun 10113

This theorem is referenced by: (None)