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Mirrors > Home > MPE Home > Th. List > grumap | Structured version Visualization version GIF version |
Description: A Grothendieck universe contains all powers of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.) |
Ref | Expression |
---|---|
grumap | ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → (𝐴 ↑𝑚 𝐵) ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1172 | . 2 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → 𝑈 ∈ Univ) | |
2 | gruxp 9944 | . . . 4 ⊢ ((𝑈 ∈ Univ ∧ 𝐵 ∈ 𝑈 ∧ 𝐴 ∈ 𝑈) → (𝐵 × 𝐴) ∈ 𝑈) | |
3 | 2 | 3com23 1162 | . . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → (𝐵 × 𝐴) ∈ 𝑈) |
4 | grupw 9932 | . . 3 ⊢ ((𝑈 ∈ Univ ∧ (𝐵 × 𝐴) ∈ 𝑈) → 𝒫 (𝐵 × 𝐴) ∈ 𝑈) | |
5 | 1, 3, 4 | syl2anc 581 | . 2 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → 𝒫 (𝐵 × 𝐴) ∈ 𝑈) |
6 | mapsspw 8158 | . . 3 ⊢ (𝐴 ↑𝑚 𝐵) ⊆ 𝒫 (𝐵 × 𝐴) | |
7 | 6 | a1i 11 | . 2 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → (𝐴 ↑𝑚 𝐵) ⊆ 𝒫 (𝐵 × 𝐴)) |
8 | gruss 9933 | . 2 ⊢ ((𝑈 ∈ Univ ∧ 𝒫 (𝐵 × 𝐴) ∈ 𝑈 ∧ (𝐴 ↑𝑚 𝐵) ⊆ 𝒫 (𝐵 × 𝐴)) → (𝐴 ↑𝑚 𝐵) ∈ 𝑈) | |
9 | 1, 5, 7, 8 | syl3anc 1496 | 1 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → (𝐴 ↑𝑚 𝐵) ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1113 ∈ wcel 2166 ⊆ wss 3798 𝒫 cpw 4378 × cxp 5340 (class class class)co 6905 ↑𝑚 cmap 8122 Univcgru 9927 |
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 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 |
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 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-fv 6131 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-1st 7428 df-2nd 7429 df-map 8124 df-pm 8125 df-gru 9928 |
This theorem is referenced by: gruixp 9946 |
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