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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 9862 | . . 3 ⊢ (𝜑 → (𝐵 ↑𝑚 𝐴) ∈ 𝑈) |
5 | 1, 4 | wunelss 9844 | . 2 ⊢ (𝜑 → (𝐵 ↑𝑚 𝐴) ⊆ 𝑈) |
6 | wunf.3 | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
7 | 2, 3 | elmapd 8135 | . . 3 ⊢ (𝜑 → (𝐹 ∈ (𝐵 ↑𝑚 𝐴) ↔ 𝐹:𝐴⟶𝐵)) |
8 | 6, 7 | mpbird 249 | . 2 ⊢ (𝜑 → 𝐹 ∈ (𝐵 ↑𝑚 𝐴)) |
9 | 5, 8 | sseldd 3827 | 1 ⊢ (𝜑 → 𝐹 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2166 ⟶wf 6118 (class class class)co 6904 ↑𝑚 cmap 8121 WUnicwun 9836 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 ax-un 7208 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-ral 3121 df-rex 3122 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-op 4403 df-uni 4658 df-iun 4741 df-br 4873 df-opab 4935 df-mpt 4952 df-tr 4975 df-id 5249 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-fv 6130 df-ov 6907 df-oprab 6908 df-mpt2 6909 df-1st 7427 df-2nd 7428 df-map 8123 df-pm 8124 df-wun 9838 |
This theorem is referenced by: wunndx 16242 wunnat 16967 catcoppccl 17109 catcfuccl 17110 catcxpccl 17199 |
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