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Theorem wunco 10750
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 10745 . . . 4 (𝜑 → dom 𝐵𝑈)
4 dmcoss 5968 . . . . 5 dom (𝐴𝐵) ⊆ dom 𝐵
54a1i 11 . . . 4 (𝜑 → dom (𝐴𝐵) ⊆ dom 𝐵)
61, 3, 5wunss 10729 . . 3 (𝜑 → dom (𝐴𝐵) ∈ 𝑈)
7 wunop.2 . . . . 5 (𝜑𝐴𝑈)
81, 7wunrn 10746 . . . 4 (𝜑 → ran 𝐴𝑈)
9 rncoss 5969 . . . . 5 ran (𝐴𝐵) ⊆ ran 𝐴
109a1i 11 . . . 4 (𝜑 → ran (𝐴𝐵) ⊆ ran 𝐴)
111, 8, 10wunss 10729 . . 3 (𝜑 → ran (𝐴𝐵) ∈ 𝑈)
121, 6, 11wunxp 10741 . 2 (𝜑 → (dom (𝐴𝐵) × ran (𝐴𝐵)) ∈ 𝑈)
13 relco 6106 . . 3 Rel (𝐴𝐵)
14 relssdmrn 6266 . . 3 (Rel (𝐴𝐵) → (𝐴𝐵) ⊆ (dom (𝐴𝐵) × ran (𝐴𝐵)))
1513, 14mp1i 13 . 2 (𝜑 → (𝐴𝐵) ⊆ (dom (𝐴𝐵) × ran (𝐴𝐵)))
161, 12, 15wunss 10729 1 (𝜑 → (𝐴𝐵) ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2099  wss 3945   × cxp 5670  dom cdm 5672  ran crn 5673  ccom 5676  Rel wrel 5677  WUnicwun 10717
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-12 2167  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-tr 5260  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-wun 10719
This theorem is referenced by: (None)
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