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Mirrors > Home > MPE Home > Th. List > gruiun | Structured version Visualization version GIF version |
Description: If 𝐵(𝑥) is a family of elements of 𝑈 and the index set 𝐴 is an element of 𝑈, then the indexed union ∪ 𝑥 ∈ 𝐴𝐵 is also an element of 𝑈, where 𝑈 is a Grothendieck universe. (Contributed by Mario Carneiro, 9-Jun-2013.) |
Ref | Expression |
---|---|
gruiun | ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2732 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
2 | 1 | fnmpt 6690 | . . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈 → (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴) |
3 | 1 | rnmptss 7124 | . . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ 𝑈) |
4 | df-f 6547 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝑈 ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ∧ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ 𝑈)) | |
5 | 2, 3, 4 | sylanbrc 583 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝑈) |
6 | gruurn 10795 | . . . . . 6 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝑈) → ∪ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝑈) | |
7 | 6 | 3expia 1121 | . . . . 5 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → ((𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝑈 → ∪ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝑈)) |
8 | 5, 7 | syl5com 31 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈 → ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → ∪ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝑈)) |
9 | dfiun3g 5963 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) | |
10 | 9 | eleq1d 2818 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈 → (∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑈 ↔ ∪ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝑈)) |
11 | 8, 10 | sylibrd 258 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈 → ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑈)) |
12 | 11 | com12 32 | . 2 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈 → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑈)) |
13 | 12 | 3impia 1117 | 1 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 ∈ wcel 2106 ∀wral 3061 ⊆ wss 3948 ∪ cuni 4908 ∪ ciun 4997 ↦ cmpt 5231 ran crn 5677 Fn wfn 6538 ⟶wf 6539 Univcgru 10787 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-ov 7414 df-oprab 7415 df-mpo 7416 df-map 8824 df-gru 10788 |
This theorem is referenced by: gruuni 10797 gruun 10803 gruixp 10806 grur1a 10816 grur1cld 43293 grucollcld 43321 |
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