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Mathbox for Rohan Ridenour |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > grucollcld | Structured version Visualization version GIF version |
Description: A Grothendieck universe contains the output of a collection operation whenever its left input is a relation on the universe, and its right input is in the universe. (Contributed by Rohan Ridenour, 11-Aug-2023.) |
Ref | Expression |
---|---|
grucollcld.1 | ⊢ (𝜑 → 𝐺 ∈ Univ) |
grucollcld.2 | ⊢ (𝜑 → 𝐹 ⊆ (𝐺 × 𝐺)) |
grucollcld.3 | ⊢ (𝜑 → 𝐴 ∈ 𝐺) |
Ref | Expression |
---|---|
grucollcld | ⊢ (𝜑 → (𝐹 Coll 𝐴) ∈ 𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfcoll2 42624 | . 2 ⊢ (𝐹 Coll 𝐴) = ∪ 𝑥 ∈ 𝐴 Scott {𝑦 ∣ 𝑥𝐹𝑦} | |
2 | grucollcld.1 | . . 3 ⊢ (𝜑 → 𝐺 ∈ Univ) | |
3 | grucollcld.3 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐺) | |
4 | simpr 486 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ Scott {𝑦 ∣ 𝑥𝐹𝑦} = ∅) → Scott {𝑦 ∣ 𝑥𝐹𝑦} = ∅) | |
5 | 2 | ad2antrr 725 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ Scott {𝑦 ∣ 𝑥𝐹𝑦} = ∅) → 𝐺 ∈ Univ) |
6 | 3 | ad2antrr 725 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ Scott {𝑦 ∣ 𝑥𝐹𝑦} = ∅) → 𝐴 ∈ 𝐺) |
7 | 5, 6 | gru0eld 42601 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ Scott {𝑦 ∣ 𝑥𝐹𝑦} = ∅) → ∅ ∈ 𝐺) |
8 | 4, 7 | eqeltrd 2834 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ Scott {𝑦 ∣ 𝑥𝐹𝑦} = ∅) → Scott {𝑦 ∣ 𝑥𝐹𝑦} ∈ 𝐺) |
9 | neq0 4309 | . . . . . . 7 ⊢ (¬ Scott {𝑦 ∣ 𝑥𝐹𝑦} = ∅ ↔ ∃𝑧 𝑧 ∈ Scott {𝑦 ∣ 𝑥𝐹𝑦}) | |
10 | 2 | ad2antrr 725 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ Scott {𝑦 ∣ 𝑥𝐹𝑦}) → 𝐺 ∈ Univ) |
11 | breq2 5113 | . . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑧 → (𝑥𝐹𝑦 ↔ 𝑥𝐹𝑧)) | |
12 | 11 | elscottab 42616 | . . . . . . . . . . . . 13 ⊢ (𝑧 ∈ Scott {𝑦 ∣ 𝑥𝐹𝑦} → 𝑥𝐹𝑧) |
13 | 12 | adantl 483 | . . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ Scott {𝑦 ∣ 𝑥𝐹𝑦}) → 𝑥𝐹𝑧) |
14 | grucollcld.2 | . . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹 ⊆ (𝐺 × 𝐺)) | |
15 | 14 | ad2antrr 725 | . . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ Scott {𝑦 ∣ 𝑥𝐹𝑦}) → 𝐹 ⊆ (𝐺 × 𝐺)) |
16 | 15 | ssbrd 5152 | . . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ Scott {𝑦 ∣ 𝑥𝐹𝑦}) → (𝑥𝐹𝑧 → 𝑥(𝐺 × 𝐺)𝑧)) |
17 | 13, 16 | mpd 15 | . . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ Scott {𝑦 ∣ 𝑥𝐹𝑦}) → 𝑥(𝐺 × 𝐺)𝑧) |
18 | brxp 5685 | . . . . . . . . . . . 12 ⊢ (𝑥(𝐺 × 𝐺)𝑧 ↔ (𝑥 ∈ 𝐺 ∧ 𝑧 ∈ 𝐺)) | |
19 | 18 | simprbi 498 | . . . . . . . . . . 11 ⊢ (𝑥(𝐺 × 𝐺)𝑧 → 𝑧 ∈ 𝐺) |
20 | 17, 19 | syl 17 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ Scott {𝑦 ∣ 𝑥𝐹𝑦}) → 𝑧 ∈ 𝐺) |
21 | simpr 486 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ Scott {𝑦 ∣ 𝑥𝐹𝑦}) → 𝑧 ∈ Scott {𝑦 ∣ 𝑥𝐹𝑦}) | |
22 | 10, 20, 21 | gruscottcld 42621 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ Scott {𝑦 ∣ 𝑥𝐹𝑦}) → Scott {𝑦 ∣ 𝑥𝐹𝑦} ∈ 𝐺) |
23 | 22 | expcom 415 | . . . . . . . 8 ⊢ (𝑧 ∈ Scott {𝑦 ∣ 𝑥𝐹𝑦} → ((𝜑 ∧ 𝑥 ∈ 𝐴) → Scott {𝑦 ∣ 𝑥𝐹𝑦} ∈ 𝐺)) |
24 | 23 | exlimiv 1934 | . . . . . . 7 ⊢ (∃𝑧 𝑧 ∈ Scott {𝑦 ∣ 𝑥𝐹𝑦} → ((𝜑 ∧ 𝑥 ∈ 𝐴) → Scott {𝑦 ∣ 𝑥𝐹𝑦} ∈ 𝐺)) |
25 | 9, 24 | sylbi 216 | . . . . . 6 ⊢ (¬ Scott {𝑦 ∣ 𝑥𝐹𝑦} = ∅ → ((𝜑 ∧ 𝑥 ∈ 𝐴) → Scott {𝑦 ∣ 𝑥𝐹𝑦} ∈ 𝐺)) |
26 | 25 | impcom 409 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ ¬ Scott {𝑦 ∣ 𝑥𝐹𝑦} = ∅) → Scott {𝑦 ∣ 𝑥𝐹𝑦} ∈ 𝐺) |
27 | 8, 26 | pm2.61dan 812 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → Scott {𝑦 ∣ 𝑥𝐹𝑦} ∈ 𝐺) |
28 | 27 | ralrimiva 3140 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 Scott {𝑦 ∣ 𝑥𝐹𝑦} ∈ 𝐺) |
29 | gruiun 10743 | . . 3 ⊢ ((𝐺 ∈ Univ ∧ 𝐴 ∈ 𝐺 ∧ ∀𝑥 ∈ 𝐴 Scott {𝑦 ∣ 𝑥𝐹𝑦} ∈ 𝐺) → ∪ 𝑥 ∈ 𝐴 Scott {𝑦 ∣ 𝑥𝐹𝑦} ∈ 𝐺) | |
30 | 2, 3, 28, 29 | syl3anc 1372 | . 2 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 Scott {𝑦 ∣ 𝑥𝐹𝑦} ∈ 𝐺) |
31 | 1, 30 | eqeltrid 2838 | 1 ⊢ (𝜑 → (𝐹 Coll 𝐴) ∈ 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 = wceq 1542 ∃wex 1782 ∈ wcel 2107 {cab 2710 ∀wral 3061 ⊆ wss 3914 ∅c0 4286 ∪ ciun 4958 class class class wbr 5109 × cxp 5635 Univcgru 10734 Scott cscott 42607 Coll ccoll 42622 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-reg 9536 ax-inf2 9585 ax-ac2 10407 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-iin 4961 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-se 5593 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-map 8773 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-tc 9681 df-r1 9708 df-rank 9709 df-card 9883 df-cf 9885 df-acn 9886 df-ac 10060 df-wina 10628 df-ina 10629 df-gru 10735 df-scott 42608 df-coll 42623 |
This theorem is referenced by: grumnudlem 42657 |
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