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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 10766 | . . 3 ⊢ (𝜑 → (𝐵 ↑m 𝐴) ∈ 𝑈) |
| 5 | 1, 4 | wunelss 10748 | . 2 ⊢ (𝜑 → (𝐵 ↑m 𝐴) ⊆ 𝑈) |
| 6 | wunf.3 | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
| 7 | 2, 3 | elmapd 8880 | . . 3 ⊢ (𝜑 → (𝐹 ∈ (𝐵 ↑m 𝐴) ↔ 𝐹:𝐴⟶𝐵)) |
| 8 | 6, 7 | mpbird 257 | . 2 ⊢ (𝜑 → 𝐹 ∈ (𝐵 ↑m 𝐴)) |
| 9 | 5, 8 | sseldd 3984 | 1 ⊢ (𝜑 → 𝐹 ∈ 𝑈) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 ⟶wf 6557 (class class class)co 7431 ↑m cmap 8866 WUnicwun 10740 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-map 8868 df-pm 8869 df-wun 10742 |
| This theorem is referenced by: wunndx 17232 wunnat 18004 catcoppccl 18162 catcfuccl 18163 catcxpccl 18252 |
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