Theorem wunco 10691

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 10686	. . . 4 ⊢ (𝜑 → dom 𝐵 ∈ 𝑈)
4		dmcoss 5951	. . . . 5 ⊢ dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵
5	4	a1i 11	. . . 4 ⊢ (𝜑 → dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵)
6	1, 3, 5	wunss 10670	. . 3 ⊢ (𝜑 → dom (𝐴 ∘ 𝐵) ∈ 𝑈)
7		wunop.2	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
8	1, 7	wunrn 10687	. . . 4 ⊢ (𝜑 → ran 𝐴 ∈ 𝑈)
9		rncoss 5953	. . . . 5 ⊢ ran (𝐴 ∘ 𝐵) ⊆ ran 𝐴
10	9	a1i 11	. . . 4 ⊢ (𝜑 → ran (𝐴 ∘ 𝐵) ⊆ ran 𝐴)
11	1, 8, 10	wunss 10670	. . 3 ⊢ (𝜑 → ran (𝐴 ∘ 𝐵) ∈ 𝑈)
12	1, 6, 11	wunxp 10682	. 2 ⊢ (𝜑 → (dom (𝐴 ∘ 𝐵) × ran (𝐴 ∘ 𝐵)) ∈ 𝑈)
13		relco 6097	. . 3 ⊢ Rel (𝐴 ∘ 𝐵)
14		relssdmrn 6256	. . 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 10670	1 ⊢ (𝜑 → (𝐴 ∘ 𝐵) ∈ 𝑈)

Syntax hints:   → wi 4   ∈ wcel 2142   ⊆ wss 3904   × cxp 5645  dom cdm 5647  ran crn 5648   ∘ ccom 5651  Rel wrel 5652  WUnicwun 10658

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-tr 5208  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-wun 10660

This theorem is referenced by: (None)