Theorem wunco 10745

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 10740	. . . 4 ⊢ (𝜑 → dom 𝐵 ∈ 𝑈)
4		dmcoss 5954	. . . . 5 ⊢ dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵
5	4	a1i 11	. . . 4 ⊢ (𝜑 → dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵)
6	1, 3, 5	wunss 10724	. . 3 ⊢ (𝜑 → dom (𝐴 ∘ 𝐵) ∈ 𝑈)
7		wunop.2	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
8	1, 7	wunrn 10741	. . . 4 ⊢ (𝜑 → ran 𝐴 ∈ 𝑈)
9		rncoss 5955	. . . . 5 ⊢ ran (𝐴 ∘ 𝐵) ⊆ ran 𝐴
10	9	a1i 11	. . . 4 ⊢ (𝜑 → ran (𝐴 ∘ 𝐵) ⊆ ran 𝐴)
11	1, 8, 10	wunss 10724	. . 3 ⊢ (𝜑 → ran (𝐴 ∘ 𝐵) ∈ 𝑈)
12	1, 6, 11	wunxp 10736	. 2 ⊢ (𝜑 → (dom (𝐴 ∘ 𝐵) × ran (𝐴 ∘ 𝐵)) ∈ 𝑈)
13		relco 6095	. . 3 ⊢ Rel (𝐴 ∘ 𝐵)
14		relssdmrn 6257	. . 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 10724	1 ⊢ (𝜑 → (𝐴 ∘ 𝐵) ∈ 𝑈)

Syntax hints:   → wi 4   ∈ wcel 2108   ⊆ wss 3926   × cxp 5652  dom cdm 5654  ran crn 5655   ∘ ccom 5658  Rel wrel 5659  WUnicwun 10712

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

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-nf 1784  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-tr 5230  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-wun 10714

This theorem is referenced by: (None)