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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 3154 | . . 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 12533 | . . . . . 6 ⊢ Base Fn V | |
8 | 6 | elexd 2762 | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ V) |
9 | funfvex 5544 | . . . . . . 7 ⊢ ((Fun Base ∧ 𝑊 ∈ dom Base) → (Base‘𝑊) ∈ V) | |
10 | 9 | funfni 5328 | . . . . . 6 ⊢ ((Base Fn V ∧ 𝑊 ∈ V) → (Base‘𝑊) ∈ V) |
11 | 7, 8, 10 | sylancr 414 | . . . . 5 ⊢ (𝜑 → (Base‘𝑊) ∈ V) |
12 | 5, 11 | eqeltrd 2264 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ V) |
13 | 12, 1 | ssexd 4155 | . . 3 ⊢ (𝜑 → 𝐴 ∈ V) |
14 | 4, 5, 6, 13 | ressbasd 12540 | . 2 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = (Base‘𝑅)) |
15 | 3, 14 | eqtr3d 2222 | 1 ⊢ (𝜑 → 𝐴 = (Base‘𝑅)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1363 ∈ wcel 2158 Vcvv 2749 ∩ cin 3140 ⊆ wss 3141 Fn wfn 5223 ‘cfv 5228 (class class class)co 5888 Basecbs 12475 ↾s cress 12476 |
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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-cnex 7915 ax-resscn 7916 ax-1re 7918 ax-addrcl 7921 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-ral 2470 df-rex 2471 df-rab 2474 df-v 2751 df-sbc 2975 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-iota 5190 df-fun 5230 df-fn 5231 df-fv 5236 df-ov 5891 df-oprab 5892 df-mpo 5893 df-inn 8933 df-ndx 12478 df-slot 12479 df-base 12481 df-sets 12482 df-iress 12483 |
This theorem is referenced by: issubmnd 12864 ress0g 12865 subgbas 13069 issubg2m 13080 ablressid 13169 rngressid 13204 ringidss 13276 ringressid 13306 unitgrpbasd 13358 islss3 13563 lsslss 13565 lsslsp 13613 2idlbas 13694 zringbas 13743 |
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