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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 10226 | . . 3 ⊢ (𝜑 → (𝐵 ↑m 𝐴) ∈ 𝑈) |
5 | 1, 4 | wunelss 10208 | . 2 ⊢ (𝜑 → (𝐵 ↑m 𝐴) ⊆ 𝑈) |
6 | wunf.3 | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
7 | 2, 3 | elmapd 8451 | . . 3 ⊢ (𝜑 → (𝐹 ∈ (𝐵 ↑m 𝐴) ↔ 𝐹:𝐴⟶𝐵)) |
8 | 6, 7 | mpbird 260 | . 2 ⊢ (𝜑 → 𝐹 ∈ (𝐵 ↑m 𝐴)) |
9 | 5, 8 | sseldd 3878 | 1 ⊢ (𝜑 → 𝐹 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 ⟶wf 6335 (class class class)co 7170 ↑m cmap 8437 WUnicwun 10200 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-fv 6347 df-ov 7173 df-oprab 7174 df-mpo 7175 df-1st 7714 df-2nd 7715 df-map 8439 df-pm 8440 df-wun 10202 |
This theorem is referenced by: wunndx 16607 wunnat 17331 catcoppccl 17484 catcfuccl 17485 catcxpccl 17573 |
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