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Mirrors > Home > MPE Home > Th. List > gruf | Structured version Visualization version GIF version |
Description: A Grothendieck universe contains all functions on its elements. (Contributed by Mario Carneiro, 10-Jun-2013.) |
Ref | Expression |
---|---|
gruf | ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐹:𝐴⟶𝑈) → 𝐹 ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp3 1135 | . . . 4 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐹:𝐴⟶𝑈) → 𝐹:𝐴⟶𝑈) | |
2 | 1 | feqmptd 6966 | . . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐹:𝐴⟶𝑈) → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))) |
3 | fvex 6909 | . . . 4 ⊢ (𝐹‘𝑥) ∈ V | |
4 | 3 | fnasrn 7154 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) = ran (𝑥 ∈ 𝐴 ↦ 〈𝑥, (𝐹‘𝑥)〉) |
5 | 2, 4 | eqtrdi 2781 | . 2 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐹:𝐴⟶𝑈) → 𝐹 = ran (𝑥 ∈ 𝐴 ↦ 〈𝑥, (𝐹‘𝑥)〉)) |
6 | simpl1 1188 | . . . . 5 ⊢ (((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐹:𝐴⟶𝑈) ∧ 𝑥 ∈ 𝐴) → 𝑈 ∈ Univ) | |
7 | gruel 10828 | . . . . . . 7 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝑈) | |
8 | 7 | 3expa 1115 | . . . . . 6 ⊢ (((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝑈) |
9 | 8 | 3adantl3 1165 | . . . . 5 ⊢ (((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐹:𝐴⟶𝑈) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝑈) |
10 | ffvelcdm 7090 | . . . . . 6 ⊢ ((𝐹:𝐴⟶𝑈 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝑈) | |
11 | 10 | 3ad2antl3 1184 | . . . . 5 ⊢ (((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐹:𝐴⟶𝑈) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝑈) |
12 | gruop 10830 | . . . . 5 ⊢ ((𝑈 ∈ Univ ∧ 𝑥 ∈ 𝑈 ∧ (𝐹‘𝑥) ∈ 𝑈) → 〈𝑥, (𝐹‘𝑥)〉 ∈ 𝑈) | |
13 | 6, 9, 11, 12 | syl3anc 1368 | . . . 4 ⊢ (((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐹:𝐴⟶𝑈) ∧ 𝑥 ∈ 𝐴) → 〈𝑥, (𝐹‘𝑥)〉 ∈ 𝑈) |
14 | 13 | fmpttd 7124 | . . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐹:𝐴⟶𝑈) → (𝑥 ∈ 𝐴 ↦ 〈𝑥, (𝐹‘𝑥)〉):𝐴⟶𝑈) |
15 | grurn 10826 | . . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ (𝑥 ∈ 𝐴 ↦ 〈𝑥, (𝐹‘𝑥)〉):𝐴⟶𝑈) → ran (𝑥 ∈ 𝐴 ↦ 〈𝑥, (𝐹‘𝑥)〉) ∈ 𝑈) | |
16 | 14, 15 | syld3an3 1406 | . 2 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐹:𝐴⟶𝑈) → ran (𝑥 ∈ 𝐴 ↦ 〈𝑥, (𝐹‘𝑥)〉) ∈ 𝑈) |
17 | 5, 16 | eqeltrd 2825 | 1 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐹:𝐴⟶𝑈) → 𝐹 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 ∈ wcel 2098 〈cop 4636 ↦ cmpt 5232 ran crn 5679 ⟶wf 6545 ‘cfv 6549 Univcgru 10815 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-map 8847 df-gru 10816 |
This theorem is referenced by: (None) |
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