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Theorem wunco 9870
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 9865 . . . 4 (𝜑 → dom 𝐵𝑈)
4 dmcoss 5618 . . . . 5 dom (𝐴𝐵) ⊆ dom 𝐵
54a1i 11 . . . 4 (𝜑 → dom (𝐴𝐵) ⊆ dom 𝐵)
61, 3, 5wunss 9849 . . 3 (𝜑 → dom (𝐴𝐵) ∈ 𝑈)
7 wunop.2 . . . . 5 (𝜑𝐴𝑈)
81, 7wunrn 9866 . . . 4 (𝜑 → ran 𝐴𝑈)
9 rncoss 5619 . . . . 5 ran (𝐴𝐵) ⊆ ran 𝐴
109a1i 11 . . . 4 (𝜑 → ran (𝐴𝐵) ⊆ ran 𝐴)
111, 8, 10wunss 9849 . . 3 (𝜑 → ran (𝐴𝐵) ∈ 𝑈)
121, 6, 11wunxp 9861 . 2 (𝜑 → (dom (𝐴𝐵) × ran (𝐴𝐵)) ∈ 𝑈)
13 relco 5874 . . 3 Rel (𝐴𝐵)
14 relssdmrn 5897 . . 3 (Rel (𝐴𝐵) → (𝐴𝐵) ⊆ (dom (𝐴𝐵) × ran (𝐴𝐵)))
1513, 14mp1i 13 . 2 (𝜑 → (𝐴𝐵) ⊆ (dom (𝐴𝐵) × ran (𝐴𝐵)))
161, 12, 15wunss 9849 1 (𝜑 → (𝐴𝐵) ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2166  wss 3798   × cxp 5340  dom cdm 5342  ran crn 5343  ccom 5346  Rel wrel 5347  WUnicwun 9837
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-tr 4976  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-wun 9839
This theorem is referenced by: (None)
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