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Theorem wunco 10773
Description: A weak universe is closed under composition. (Contributed by Mario Carneiro, 12-Jan-2017.)
Hypotheses
Ref Expression
wun0.1 (𝜑𝑈 ∈ WUni)
wunop.2 (𝜑𝐴𝑈)
wunco.3 (𝜑𝐵𝑈)
Assertion
Ref Expression
wunco (𝜑 → (𝐴𝐵) ∈ 𝑈)

Proof of Theorem wunco
StepHypRef Expression
1 wun0.1 . 2 (𝜑𝑈 ∈ WUni)
2 wunco.3 . . . . 5 (𝜑𝐵𝑈)
31, 2wundm 10768 . . . 4 (𝜑 → dom 𝐵𝑈)
4 dmcoss 5985 . . . . 5 dom (𝐴𝐵) ⊆ dom 𝐵
54a1i 11 . . . 4 (𝜑 → dom (𝐴𝐵) ⊆ dom 𝐵)
61, 3, 5wunss 10752 . . 3 (𝜑 → dom (𝐴𝐵) ∈ 𝑈)
7 wunop.2 . . . . 5 (𝜑𝐴𝑈)
81, 7wunrn 10769 . . . 4 (𝜑 → ran 𝐴𝑈)
9 rncoss 5986 . . . . 5 ran (𝐴𝐵) ⊆ ran 𝐴
109a1i 11 . . . 4 (𝜑 → ran (𝐴𝐵) ⊆ ran 𝐴)
111, 8, 10wunss 10752 . . 3 (𝜑 → ran (𝐴𝐵) ∈ 𝑈)
121, 6, 11wunxp 10764 . 2 (𝜑 → (dom (𝐴𝐵) × ran (𝐴𝐵)) ∈ 𝑈)
13 relco 6126 . . 3 Rel (𝐴𝐵)
14 relssdmrn 6288 . . 3 (Rel (𝐴𝐵) → (𝐴𝐵) ⊆ (dom (𝐴𝐵) × ran (𝐴𝐵)))
1513, 14mp1i 13 . 2 (𝜑 → (𝐴𝐵) ⊆ (dom (𝐴𝐵) × ran (𝐴𝐵)))
161, 12, 15wunss 10752 1 (𝜑 → (𝐴𝐵) ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  wss 3951   × cxp 5683  dom cdm 5685  ran crn 5686  ccom 5689  Rel wrel 5690  WUnicwun 10740
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 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-wun 10742
This theorem is referenced by: (None)
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