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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 9823 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 8022 | . . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → (𝐹 ∈ (𝑈 ↑𝑚 𝐴) ↔ 𝐹:𝐴⟶𝑈)) | |
2 | elgrug 9816 | . . . . . . 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 256 | . . . . . 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 483 | . . . . 5 ⊢ (𝑈 ∈ Univ → ∀𝑥 ∈ 𝑈 (𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈 ↑𝑚 𝑥)∪ ran 𝑦 ∈ 𝑈)) |
5 | rneq 5489 | . . . . . . . . . 10 ⊢ (𝑦 = 𝐹 → ran 𝑦 = ran 𝐹) | |
6 | 5 | unieqd 4584 | . . . . . . . . 9 ⊢ (𝑦 = 𝐹 → ∪ ran 𝑦 = ∪ ran 𝐹) |
7 | 6 | eleq1d 2835 | . . . . . . . 8 ⊢ (𝑦 = 𝐹 → (∪ ran 𝑦 ∈ 𝑈 ↔ ∪ ran 𝐹 ∈ 𝑈)) |
8 | 7 | rspccv 3457 | . . . . . . 7 ⊢ (∀𝑦 ∈ (𝑈 ↑𝑚 𝑥)∪ ran 𝑦 ∈ 𝑈 → (𝐹 ∈ (𝑈 ↑𝑚 𝑥) → ∪ ran 𝐹 ∈ 𝑈)) |
9 | 8 | 3ad2ant3 1129 | . . . . . 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 3101 | . . . . 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 6801 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑈 ↑𝑚 𝑥) = (𝑈 ↑𝑚 𝐴)) | |
12 | 11 | eleq2d 2836 | . . . . . . 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 3457 | . . . . 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 393 | . . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → (𝐹 ∈ (𝑈 ↑𝑚 𝐴) → ∪ ran 𝐹 ∈ 𝑈)) |
17 | 1, 16 | sylbird 250 | . 2 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → (𝐹:𝐴⟶𝑈 → ∪ ran 𝐹 ∈ 𝑈)) |
18 | 17 | 3impia 1109 | 1 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐹:𝐴⟶𝑈) → ∪ ran 𝐹 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 ∧ w3a 1071 = wceq 1631 ∈ wcel 2145 ∀wral 3061 𝒫 cpw 4297 {cpr 4318 ∪ cuni 4574 Tr wtr 4886 ran crn 5250 ⟶wf 6027 (class class class)co 6793 ↑𝑚 cmap 8009 Univcgru 9814 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-br 4787 df-opab 4847 df-tr 4887 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-fv 6039 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-map 8011 df-gru 9815 |
This theorem is referenced by: gruiun 9823 grurn 9825 intgru 9838 |
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