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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 10790 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 1137 | . 2 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐹:𝐴⟶𝑈) → 𝑈 ∈ Univ) | |
2 | gruurn 10789 | . . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐹:𝐴⟶𝑈) → ∪ ran 𝐹 ∈ 𝑈) | |
3 | grupw 10786 | . . 3 ⊢ ((𝑈 ∈ Univ ∧ ∪ ran 𝐹 ∈ 𝑈) → 𝒫 ∪ ran 𝐹 ∈ 𝑈) | |
4 | 1, 2, 3 | syl2anc 585 | . 2 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐹:𝐴⟶𝑈) → 𝒫 ∪ ran 𝐹 ∈ 𝑈) |
5 | pwuni 4948 | . . 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 10787 | . 2 ⊢ ((𝑈 ∈ Univ ∧ 𝒫 ∪ ran 𝐹 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝒫 ∪ ran 𝐹) → ran 𝐹 ∈ 𝑈) | |
8 | 1, 4, 6, 7 | syl3anc 1372 | 1 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐹:𝐴⟶𝑈) → ran 𝐹 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1088 ∈ wcel 2107 ⊆ wss 3947 𝒫 cpw 4601 ∪ cuni 4907 ran crn 5676 ⟶wf 6536 Univcgru 10781 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-tr 5265 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-fv 6548 df-ov 7407 df-oprab 7408 df-mpo 7409 df-map 8818 df-gru 10782 |
This theorem is referenced by: gruima 10793 gruf 10802 gruen 10803 |
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