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Mirrors > Home > MPE Home > Th. List > grurn | Structured version Visualization version GIF version |
Description: A Grothendieck universe contains the range of any function which takes values in the universe (see gruiun 10210 for a more intuitive version). (Contributed by Mario Carneiro, 9-Jun-2013.) |
Ref | Expression |
---|---|
grurn | ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐹:𝐴⟶𝑈) → ran 𝐹 ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1133 | . 2 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐹:𝐴⟶𝑈) → 𝑈 ∈ Univ) | |
2 | gruurn 10209 | . . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐹:𝐴⟶𝑈) → ∪ ran 𝐹 ∈ 𝑈) | |
3 | grupw 10206 | . . 3 ⊢ ((𝑈 ∈ Univ ∧ ∪ ran 𝐹 ∈ 𝑈) → 𝒫 ∪ ran 𝐹 ∈ 𝑈) | |
4 | 1, 2, 3 | syl2anc 587 | . 2 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐹:𝐴⟶𝑈) → 𝒫 ∪ ran 𝐹 ∈ 𝑈) |
5 | pwuni 4837 | . . 3 ⊢ ran 𝐹 ⊆ 𝒫 ∪ ran 𝐹 | |
6 | 5 | a1i 11 | . 2 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐹:𝐴⟶𝑈) → ran 𝐹 ⊆ 𝒫 ∪ ran 𝐹) |
7 | gruss 10207 | . 2 ⊢ ((𝑈 ∈ Univ ∧ 𝒫 ∪ ran 𝐹 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝒫 ∪ ran 𝐹) → ran 𝐹 ∈ 𝑈) | |
8 | 1, 4, 6, 7 | syl3anc 1368 | 1 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐹:𝐴⟶𝑈) → ran 𝐹 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 ∈ wcel 2111 ⊆ wss 3881 𝒫 cpw 4497 ∪ cuni 4800 ran crn 5520 ⟶wf 6320 Univcgru 10201 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-tr 5137 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-map 8391 df-gru 10202 |
This theorem is referenced by: gruima 10213 gruf 10222 gruen 10223 |
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