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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 10209 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 8408 | . . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → (𝐹 ∈ (𝑈 ↑m 𝐴) ↔ 𝐹:𝐴⟶𝑈)) | |
2 | elgrug 10202 | . . . . . . 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 268 | . . . . . 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 496 | . . . . 5 ⊢ (𝑈 ∈ Univ → ∀𝑥 ∈ 𝑈 (𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑈)) |
5 | rneq 5799 | . . . . . . . . . 10 ⊢ (𝑦 = 𝐹 → ran 𝑦 = ran 𝐹) | |
6 | 5 | unieqd 4840 | . . . . . . . . 9 ⊢ (𝑦 = 𝐹 → ∪ ran 𝑦 = ∪ ran 𝐹) |
7 | 6 | eleq1d 2894 | . . . . . . . 8 ⊢ (𝑦 = 𝐹 → (∪ ran 𝑦 ∈ 𝑈 ↔ ∪ ran 𝐹 ∈ 𝑈)) |
8 | 7 | rspccv 3617 | . . . . . . 7 ⊢ (∀𝑦 ∈ (𝑈 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑈 → (𝐹 ∈ (𝑈 ↑m 𝑥) → ∪ ran 𝐹 ∈ 𝑈)) |
9 | 8 | 3ad2ant3 1127 | . . . . . 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 3157 | . . . . 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 7153 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑈 ↑m 𝑥) = (𝑈 ↑m 𝐴)) | |
12 | 11 | eleq2d 2895 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝐹 ∈ (𝑈 ↑m 𝑥) ↔ 𝐹 ∈ (𝑈 ↑m 𝐴))) |
13 | 12 | imbi1d 343 | . . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝐹 ∈ (𝑈 ↑m 𝑥) → ∪ ran 𝐹 ∈ 𝑈) ↔ (𝐹 ∈ (𝑈 ↑m 𝐴) → ∪ ran 𝐹 ∈ 𝑈))) |
14 | 13 | rspccv 3617 | . . . . 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 407 | . . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → (𝐹 ∈ (𝑈 ↑m 𝐴) → ∪ ran 𝐹 ∈ 𝑈)) |
17 | 1, 16 | sylbird 261 | . 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 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ∀wral 3135 𝒫 cpw 4535 {cpr 4559 ∪ cuni 4830 Tr wtr 5163 ran crn 5549 ⟶wf 6344 (class class class)co 7145 ↑m cmap 8395 Univcgru 10200 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-tr 5164 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-map 8397 df-gru 10201 |
This theorem is referenced by: gruiun 10209 grurn 10211 intgru 10224 |
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