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Mirrors > Home > MPE Home > Th. List > wunco | Structured version Visualization version GIF version |
Description: A weak universe is closed under composition. (Contributed by Mario Carneiro, 12-Jan-2017.) |
Ref | Expression |
---|---|
wun0.1 | ⊢ (𝜑 → 𝑈 ∈ WUni) |
wunop.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑈) |
wunco.3 | ⊢ (𝜑 → 𝐵 ∈ 𝑈) |
Ref | Expression |
---|---|
wunco | ⊢ (𝜑 → (𝐴 ∘ 𝐵) ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wun0.1 | . 2 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
2 | wunco.3 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑈) | |
3 | 1, 2 | wundm 10144 | . . . 4 ⊢ (𝜑 → dom 𝐵 ∈ 𝑈) |
4 | dmcoss 5836 | . . . . 5 ⊢ dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵 | |
5 | 4 | a1i 11 | . . . 4 ⊢ (𝜑 → dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵) |
6 | 1, 3, 5 | wunss 10128 | . . 3 ⊢ (𝜑 → dom (𝐴 ∘ 𝐵) ∈ 𝑈) |
7 | wunop.2 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
8 | 1, 7 | wunrn 10145 | . . . 4 ⊢ (𝜑 → ran 𝐴 ∈ 𝑈) |
9 | rncoss 5837 | . . . . 5 ⊢ ran (𝐴 ∘ 𝐵) ⊆ ran 𝐴 | |
10 | 9 | a1i 11 | . . . 4 ⊢ (𝜑 → ran (𝐴 ∘ 𝐵) ⊆ ran 𝐴) |
11 | 1, 8, 10 | wunss 10128 | . . 3 ⊢ (𝜑 → ran (𝐴 ∘ 𝐵) ∈ 𝑈) |
12 | 1, 6, 11 | wunxp 10140 | . 2 ⊢ (𝜑 → (dom (𝐴 ∘ 𝐵) × ran (𝐴 ∘ 𝐵)) ∈ 𝑈) |
13 | relco 6091 | . . 3 ⊢ Rel (𝐴 ∘ 𝐵) | |
14 | relssdmrn 6115 | . . 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 10128 | 1 ⊢ (𝜑 → (𝐴 ∘ 𝐵) ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 ⊆ wss 3935 × cxp 5547 dom cdm 5549 ran crn 5550 ∘ ccom 5553 Rel wrel 5554 WUnicwun 10116 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-tr 5165 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-wun 10118 |
This theorem is referenced by: (None) |
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