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Theorem wunco 9500
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 9495 . . . 4 (𝜑 → dom 𝐵𝑈)
4 dmcoss 5349 . . . . 5 dom (𝐴𝐵) ⊆ dom 𝐵
54a1i 11 . . . 4 (𝜑 → dom (𝐴𝐵) ⊆ dom 𝐵)
61, 3, 5wunss 9479 . . 3 (𝜑 → dom (𝐴𝐵) ∈ 𝑈)
7 wunop.2 . . . . 5 (𝜑𝐴𝑈)
81, 7wunrn 9496 . . . 4 (𝜑 → ran 𝐴𝑈)
9 rncoss 5350 . . . . 5 ran (𝐴𝐵) ⊆ ran 𝐴
109a1i 11 . . . 4 (𝜑 → ran (𝐴𝐵) ⊆ ran 𝐴)
111, 8, 10wunss 9479 . . 3 (𝜑 → ran (𝐴𝐵) ∈ 𝑈)
121, 6, 11wunxp 9491 . 2 (𝜑 → (dom (𝐴𝐵) × ran (𝐴𝐵)) ∈ 𝑈)
13 relco 5595 . . 3 Rel (𝐴𝐵)
14 relssdmrn 5618 . . 3 (Rel (𝐴𝐵) → (𝐴𝐵) ⊆ (dom (𝐴𝐵) × ran (𝐴𝐵)))
1513, 14mp1i 13 . 2 (𝜑 → (𝐴𝐵) ⊆ (dom (𝐴𝐵) × ran (𝐴𝐵)))
161, 12, 15wunss 9479 1 (𝜑 → (𝐴𝐵) ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1992  wss 3560   × cxp 5077  dom cdm 5079  ran crn 5080  ccom 5083  Rel wrel 5084  WUnicwun 9467
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-tr 4718  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-wun 9469
This theorem is referenced by: (None)
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