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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 10150 | . . 3 ⊢ (𝜑 → (𝐵 ↑m 𝐴) ∈ 𝑈) |
5 | 1, 4 | wunelss 10132 | . 2 ⊢ (𝜑 → (𝐵 ↑m 𝐴) ⊆ 𝑈) |
6 | wunf.3 | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
7 | 2, 3 | elmapd 8422 | . . 3 ⊢ (𝜑 → (𝐹 ∈ (𝐵 ↑m 𝐴) ↔ 𝐹:𝐴⟶𝐵)) |
8 | 6, 7 | mpbird 259 | . 2 ⊢ (𝜑 → 𝐹 ∈ (𝐵 ↑m 𝐴)) |
9 | 5, 8 | sseldd 3970 | 1 ⊢ (𝜑 → 𝐹 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 ⟶wf 6353 (class class class)co 7158 ↑m cmap 8408 WUnicwun 10124 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-map 8410 df-pm 8411 df-wun 10126 |
This theorem is referenced by: wunndx 16506 wunnat 17228 catcoppccl 17370 catcfuccl 17371 catcxpccl 17459 |
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