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Mirrors > Home > ILE Home > Th. List > qusex | GIF version |
Description: Existence of a quotient structure. (Contributed by Jim Kingdon, 25-Apr-2025.) |
Ref | Expression |
---|---|
qusex | ⊢ ((𝑅 ∈ 𝑉 ∧ ∼ ∈ 𝑊) → (𝑅 /s ∼ ) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2763 | . . . 4 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V) | |
2 | 1 | adantr 276 | . . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ ∼ ∈ 𝑊) → 𝑅 ∈ V) |
3 | elex 2763 | . . . 4 ⊢ ( ∼ ∈ 𝑊 → ∼ ∈ V) | |
4 | 3 | adantl 277 | . . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ ∼ ∈ 𝑊) → ∼ ∈ V) |
5 | basfn 12565 | . . . . . 6 ⊢ Base Fn V | |
6 | funfvex 5548 | . . . . . . 7 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V) | |
7 | 6 | funfni 5332 | . . . . . 6 ⊢ ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V) |
8 | 5, 2, 7 | sylancr 414 | . . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ ∼ ∈ 𝑊) → (Base‘𝑅) ∈ V) |
9 | 8 | mptexd 5760 | . . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ ∼ ∈ 𝑊) → (𝑥 ∈ (Base‘𝑅) ↦ [𝑥] ∼ ) ∈ V) |
10 | simpl 109 | . . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ ∼ ∈ 𝑊) → 𝑅 ∈ 𝑉) | |
11 | imasex 12775 | . . . 4 ⊢ (((𝑥 ∈ (Base‘𝑅) ↦ [𝑥] ∼ ) ∈ V ∧ 𝑅 ∈ 𝑉) → ((𝑥 ∈ (Base‘𝑅) ↦ [𝑥] ∼ ) “s 𝑅) ∈ V) | |
12 | 9, 10, 11 | syl2anc 411 | . . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ ∼ ∈ 𝑊) → ((𝑥 ∈ (Base‘𝑅) ↦ [𝑥] ∼ ) “s 𝑅) ∈ V) |
13 | fveq2 5531 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅)) | |
14 | 13 | mpteq1d 4103 | . . . . 5 ⊢ (𝑟 = 𝑅 → (𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) = (𝑥 ∈ (Base‘𝑅) ↦ [𝑥]𝑒)) |
15 | id 19 | . . . . 5 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅) | |
16 | 14, 15 | oveq12d 5910 | . . . 4 ⊢ (𝑟 = 𝑅 → ((𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) “s 𝑟) = ((𝑥 ∈ (Base‘𝑅) ↦ [𝑥]𝑒) “s 𝑅)) |
17 | eceq2 6591 | . . . . . 6 ⊢ (𝑒 = ∼ → [𝑥]𝑒 = [𝑥] ∼ ) | |
18 | 17 | mpteq2dv 4109 | . . . . 5 ⊢ (𝑒 = ∼ → (𝑥 ∈ (Base‘𝑅) ↦ [𝑥]𝑒) = (𝑥 ∈ (Base‘𝑅) ↦ [𝑥] ∼ )) |
19 | 18 | oveq1d 5907 | . . . 4 ⊢ (𝑒 = ∼ → ((𝑥 ∈ (Base‘𝑅) ↦ [𝑥]𝑒) “s 𝑅) = ((𝑥 ∈ (Base‘𝑅) ↦ [𝑥] ∼ ) “s 𝑅)) |
20 | df-qus 12773 | . . . 4 ⊢ /s = (𝑟 ∈ V, 𝑒 ∈ V ↦ ((𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) “s 𝑟)) | |
21 | 16, 19, 20 | ovmpog 6027 | . . 3 ⊢ ((𝑅 ∈ V ∧ ∼ ∈ V ∧ ((𝑥 ∈ (Base‘𝑅) ↦ [𝑥] ∼ ) “s 𝑅) ∈ V) → (𝑅 /s ∼ ) = ((𝑥 ∈ (Base‘𝑅) ↦ [𝑥] ∼ ) “s 𝑅)) |
22 | 2, 4, 12, 21 | syl3anc 1249 | . 2 ⊢ ((𝑅 ∈ 𝑉 ∧ ∼ ∈ 𝑊) → (𝑅 /s ∼ ) = ((𝑥 ∈ (Base‘𝑅) ↦ [𝑥] ∼ ) “s 𝑅)) |
23 | 22, 12 | eqeltrd 2266 | 1 ⊢ ((𝑅 ∈ 𝑉 ∧ ∼ ∈ 𝑊) → (𝑅 /s ∼ ) ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2160 Vcvv 2752 ↦ cmpt 4079 Fn wfn 5227 ‘cfv 5232 (class class class)co 5892 [cec 6552 Basecbs 12507 “s cimas 12769 /s cqus 12770 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-cnex 7927 ax-resscn 7928 ax-1re 7930 ax-addrcl 7933 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-tp 3615 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5234 df-fn 5235 df-f 5236 df-f1 5237 df-fo 5238 df-f1o 5239 df-fv 5240 df-ov 5895 df-oprab 5896 df-mpo 5897 df-ec 6556 df-inn 8945 df-2 9003 df-3 9004 df-ndx 12510 df-slot 12511 df-base 12513 df-plusg 12595 df-mulr 12596 df-iimas 12772 df-qus 12773 |
This theorem is referenced by: znval 13925 znle 13926 znbaslemnn 13928 |
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