![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 10745 | . . . 4 ⊢ (𝜑 → dom 𝐵 ∈ 𝑈) |
4 | dmcoss 5968 | . . . . 5 ⊢ dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵 | |
5 | 4 | a1i 11 | . . . 4 ⊢ (𝜑 → dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵) |
6 | 1, 3, 5 | wunss 10729 | . . 3 ⊢ (𝜑 → dom (𝐴 ∘ 𝐵) ∈ 𝑈) |
7 | wunop.2 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
8 | 1, 7 | wunrn 10746 | . . . 4 ⊢ (𝜑 → ran 𝐴 ∈ 𝑈) |
9 | rncoss 5969 | . . . . 5 ⊢ ran (𝐴 ∘ 𝐵) ⊆ ran 𝐴 | |
10 | 9 | a1i 11 | . . . 4 ⊢ (𝜑 → ran (𝐴 ∘ 𝐵) ⊆ ran 𝐴) |
11 | 1, 8, 10 | wunss 10729 | . . 3 ⊢ (𝜑 → ran (𝐴 ∘ 𝐵) ∈ 𝑈) |
12 | 1, 6, 11 | wunxp 10741 | . 2 ⊢ (𝜑 → (dom (𝐴 ∘ 𝐵) × ran (𝐴 ∘ 𝐵)) ∈ 𝑈) |
13 | relco 6106 | . . 3 ⊢ Rel (𝐴 ∘ 𝐵) | |
14 | relssdmrn 6266 | . . 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 10729 | 1 ⊢ (𝜑 → (𝐴 ∘ 𝐵) ∈ 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) |
Copyright terms: Public domain | W3C validator |