![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 10067 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 8269 | . . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → (𝐹 ∈ (𝑈 ↑𝑚 𝐴) ↔ 𝐹:𝐴⟶𝑈)) | |
2 | elgrug 10060 | . . . . . . 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 5688 | . . . . . . . . . 10 ⊢ (𝑦 = 𝐹 → ran 𝑦 = ran 𝐹) | |
6 | 5 | unieqd 4755 | . . . . . . . . 9 ⊢ (𝑦 = 𝐹 → ∪ ran 𝑦 = ∪ ran 𝐹) |
7 | 6 | eleq1d 2867 | . . . . . . . 8 ⊢ (𝑦 = 𝐹 → (∪ ran 𝑦 ∈ 𝑈 ↔ ∪ ran 𝐹 ∈ 𝑈)) |
8 | 7 | rspccv 3556 | . . . . . . 7 ⊢ (∀𝑦 ∈ (𝑈 ↑𝑚 𝑥)∪ ran 𝑦 ∈ 𝑈 → (𝐹 ∈ (𝑈 ↑𝑚 𝑥) → ∪ ran 𝐹 ∈ 𝑈)) |
9 | 8 | 3ad2ant3 1128 | . . . . . 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 3127 | . . . . 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 7024 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑈 ↑𝑚 𝑥) = (𝑈 ↑𝑚 𝐴)) | |
12 | 11 | eleq2d 2868 | . . . . . . 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 3556 | . . . . 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 1110 | 1 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐹:𝐴⟶𝑈) → ∪ ran 𝐹 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1080 = wceq 1522 ∈ wcel 2081 ∀wral 3105 𝒫 cpw 4453 {cpr 4474 ∪ cuni 4745 Tr wtr 5063 ran crn 5444 ⟶wf 6221 (class class class)co 7016 ↑𝑚 cmap 8256 Univcgru 10058 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ral 3110 df-rex 3111 df-rab 3114 df-v 3439 df-sbc 3707 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-op 4479 df-uni 4746 df-br 4963 df-opab 5025 df-tr 5064 df-id 5348 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-fv 6233 df-ov 7019 df-oprab 7020 df-mpo 7021 df-map 8258 df-gru 10059 |
This theorem is referenced by: gruiun 10067 grurn 10069 intgru 10082 |
Copyright terms: Public domain | W3C validator |