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Mirrors > Home > MPE Home > Th. List > wunf | Structured version Visualization version GIF version |
Description: A weak universe is closed under functions with known domain and codomain. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Ref | Expression |
---|---|
wun0.1 | ⊢ (𝜑 → 𝑈 ∈ WUni) |
wunop.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑈) |
wunop.3 | ⊢ (𝜑 → 𝐵 ∈ 𝑈) |
wunf.3 | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
Ref | Expression |
---|---|
wunf | ⊢ (𝜑 → 𝐹 ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wun0.1 | . . 3 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
2 | wunop.3 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑈) | |
3 | wunop.2 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
4 | 1, 2, 3 | wunmap 10763 | . . 3 ⊢ (𝜑 → (𝐵 ↑m 𝐴) ∈ 𝑈) |
5 | 1, 4 | wunelss 10745 | . 2 ⊢ (𝜑 → (𝐵 ↑m 𝐴) ⊆ 𝑈) |
6 | wunf.3 | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
7 | 2, 3 | elmapd 8878 | . . 3 ⊢ (𝜑 → (𝐹 ∈ (𝐵 ↑m 𝐴) ↔ 𝐹:𝐴⟶𝐵)) |
8 | 6, 7 | mpbird 257 | . 2 ⊢ (𝜑 → 𝐹 ∈ (𝐵 ↑m 𝐴)) |
9 | 5, 8 | sseldd 3995 | 1 ⊢ (𝜑 → 𝐹 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ⟶wf 6558 (class class class)co 7430 ↑m cmap 8864 WUnicwun 10737 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 df-1st 8012 df-2nd 8013 df-map 8866 df-pm 8867 df-wun 10739 |
This theorem is referenced by: wunndx 17228 wunnat 18010 wunnatOLD 18011 catcoppccl 18170 catcoppcclOLD 18171 catcfuccl 18172 catcfucclOLD 18173 catcxpccl 18262 catcxpcclOLD 18263 |
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