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Mirrors > Home > MPE Home > Th. List > gruurn | Structured version Visualization version GIF version |
Description: A Grothendieck universe contains the range of any function which takes values in the universe (see gruiun 9500 for a more intuitive version). (Contributed by Mario Carneiro, 9-Jun-2013.) |
Ref | Expression |
---|---|
gruurn | ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐹:𝐴⟶𝑈) → ∪ ran 𝐹 ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elmapg 7757 | . . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → (𝐹 ∈ (𝑈 ↑𝑚 𝐴) ↔ 𝐹:𝐴⟶𝑈)) | |
2 | elgrug 9493 | . . . . . . 7 ⊢ (𝑈 ∈ Univ → (𝑈 ∈ Univ ↔ (Tr 𝑈 ∧ ∀𝑥 ∈ 𝑈 (𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈 ↑𝑚 𝑥)∪ ran 𝑦 ∈ 𝑈)))) | |
3 | 2 | ibi 255 | . . . . . 6 ⊢ (𝑈 ∈ Univ → (Tr 𝑈 ∧ ∀𝑥 ∈ 𝑈 (𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈 ↑𝑚 𝑥)∪ ran 𝑦 ∈ 𝑈))) |
4 | 3 | simprd 478 | . . . . 5 ⊢ (𝑈 ∈ Univ → ∀𝑥 ∈ 𝑈 (𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈 ↑𝑚 𝑥)∪ ran 𝑦 ∈ 𝑈)) |
5 | rneq 5272 | . . . . . . . . . 10 ⊢ (𝑦 = 𝐹 → ran 𝑦 = ran 𝐹) | |
6 | 5 | unieqd 4382 | . . . . . . . . 9 ⊢ (𝑦 = 𝐹 → ∪ ran 𝑦 = ∪ ran 𝐹) |
7 | 6 | eleq1d 2672 | . . . . . . . 8 ⊢ (𝑦 = 𝐹 → (∪ ran 𝑦 ∈ 𝑈 ↔ ∪ ran 𝐹 ∈ 𝑈)) |
8 | 7 | rspccv 3279 | . . . . . . 7 ⊢ (∀𝑦 ∈ (𝑈 ↑𝑚 𝑥)∪ ran 𝑦 ∈ 𝑈 → (𝐹 ∈ (𝑈 ↑𝑚 𝑥) → ∪ ran 𝐹 ∈ 𝑈)) |
9 | 8 | 3ad2ant3 1077 | . . . . . 6 ⊢ ((𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈 ↑𝑚 𝑥)∪ ran 𝑦 ∈ 𝑈) → (𝐹 ∈ (𝑈 ↑𝑚 𝑥) → ∪ ran 𝐹 ∈ 𝑈)) |
10 | 9 | ralimi 2936 | . . . . 5 ⊢ (∀𝑥 ∈ 𝑈 (𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈 ↑𝑚 𝑥)∪ ran 𝑦 ∈ 𝑈) → ∀𝑥 ∈ 𝑈 (𝐹 ∈ (𝑈 ↑𝑚 𝑥) → ∪ ran 𝐹 ∈ 𝑈)) |
11 | oveq2 6557 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑈 ↑𝑚 𝑥) = (𝑈 ↑𝑚 𝐴)) | |
12 | 11 | eleq2d 2673 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝐹 ∈ (𝑈 ↑𝑚 𝑥) ↔ 𝐹 ∈ (𝑈 ↑𝑚 𝐴))) |
13 | 12 | imbi1d 330 | . . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝐹 ∈ (𝑈 ↑𝑚 𝑥) → ∪ ran 𝐹 ∈ 𝑈) ↔ (𝐹 ∈ (𝑈 ↑𝑚 𝐴) → ∪ ran 𝐹 ∈ 𝑈))) |
14 | 13 | rspccv 3279 | . . . . 5 ⊢ (∀𝑥 ∈ 𝑈 (𝐹 ∈ (𝑈 ↑𝑚 𝑥) → ∪ ran 𝐹 ∈ 𝑈) → (𝐴 ∈ 𝑈 → (𝐹 ∈ (𝑈 ↑𝑚 𝐴) → ∪ ran 𝐹 ∈ 𝑈))) |
15 | 4, 10, 14 | 3syl 18 | . . . 4 ⊢ (𝑈 ∈ Univ → (𝐴 ∈ 𝑈 → (𝐹 ∈ (𝑈 ↑𝑚 𝐴) → ∪ ran 𝐹 ∈ 𝑈))) |
16 | 15 | imp 444 | . . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → (𝐹 ∈ (𝑈 ↑𝑚 𝐴) → ∪ ran 𝐹 ∈ 𝑈)) |
17 | 1, 16 | sylbird 249 | . 2 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → (𝐹:𝐴⟶𝑈 → ∪ ran 𝐹 ∈ 𝑈)) |
18 | 17 | 3impia 1253 | 1 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐹:𝐴⟶𝑈) → ∪ ran 𝐹 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 𝒫 cpw 4108 {cpr 4127 ∪ cuni 4372 Tr wtr 4680 ran crn 5039 ⟶wf 5800 (class class class)co 6549 ↑𝑚 cmap 7744 Univcgru 9491 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-tr 4681 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-map 7746 df-gru 9492 |
This theorem is referenced by: gruiun 9500 grurn 9502 intgru 9515 |
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