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Mirrors > Home > ILE Home > Th. List > ressbas2d | GIF version |
Description: Base set of a structure restriction. (Contributed by Mario Carneiro, 2-Dec-2014.) |
Ref | Expression |
---|---|
ressbasd.r | ⊢ (𝜑 → 𝑅 = (𝑊 ↾s 𝐴)) |
ressbasd.b | ⊢ (𝜑 → 𝐵 = (Base‘𝑊)) |
ressbasd.w | ⊢ (𝜑 → 𝑊 ∈ 𝑋) |
ressbas2d.ss | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
Ref | Expression |
---|---|
ressbas2d | ⊢ (𝜑 → 𝐴 = (Base‘𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ressbas2d.ss | . . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
2 | df-ss 3157 | . . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴) | |
3 | 1, 2 | sylib 122 | . 2 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = 𝐴) |
4 | ressbasd.r | . . 3 ⊢ (𝜑 → 𝑅 = (𝑊 ↾s 𝐴)) | |
5 | ressbasd.b | . . 3 ⊢ (𝜑 → 𝐵 = (Base‘𝑊)) | |
6 | ressbasd.w | . . 3 ⊢ (𝜑 → 𝑊 ∈ 𝑋) | |
7 | basfn 12544 | . . . . . 6 ⊢ Base Fn V | |
8 | 6 | elexd 2765 | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ V) |
9 | funfvex 5547 | . . . . . . 7 ⊢ ((Fun Base ∧ 𝑊 ∈ dom Base) → (Base‘𝑊) ∈ V) | |
10 | 9 | funfni 5331 | . . . . . 6 ⊢ ((Base Fn V ∧ 𝑊 ∈ V) → (Base‘𝑊) ∈ V) |
11 | 7, 8, 10 | sylancr 414 | . . . . 5 ⊢ (𝜑 → (Base‘𝑊) ∈ V) |
12 | 5, 11 | eqeltrd 2266 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ V) |
13 | 12, 1 | ssexd 4158 | . . 3 ⊢ (𝜑 → 𝐴 ∈ V) |
14 | 4, 5, 6, 13 | ressbasd 12551 | . 2 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = (Base‘𝑅)) |
15 | 3, 14 | eqtr3d 2224 | 1 ⊢ (𝜑 → 𝐴 = (Base‘𝑅)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 ∈ wcel 2160 Vcvv 2752 ∩ cin 3143 ⊆ wss 3144 Fn wfn 5226 ‘cfv 5231 (class class class)co 5891 Basecbs 12486 ↾s cress 12487 |
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-sep 4136 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-cnex 7921 ax-resscn 7922 ax-1re 7924 ax-addrcl 7927 |
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-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 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-iota 5193 df-fun 5233 df-fn 5234 df-fv 5239 df-ov 5894 df-oprab 5895 df-mpo 5896 df-inn 8939 df-ndx 12489 df-slot 12490 df-base 12492 df-sets 12493 df-iress 12494 |
This theorem is referenced by: issubmnd 12875 ress0g 12876 submbas 12905 resmhm 12911 subgbas 13089 issubg2m 13100 resghm 13166 ablressid 13239 rngressid 13275 ringidss 13350 ringressid 13380 unitgrpbasd 13432 islss3 13662 lsslss 13664 lsslsp 13712 2idlbas 13797 zringbas 13862 |
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