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Mirrors > Home > MPE Home > Th. List > ressbasss | Structured version Visualization version GIF version |
Description: The base set of a restriction is a subset of the base set of the original structure. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) |
Ref | Expression |
---|---|
ressbas.r | ⊢ 𝑅 = (𝑊 ↾s 𝐴) |
ressbas.b | ⊢ 𝐵 = (Base‘𝑊) |
Ref | Expression |
---|---|
ressbasss | ⊢ (Base‘𝑅) ⊆ 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ressbas.r | . . . 4 ⊢ 𝑅 = (𝑊 ↾s 𝐴) | |
2 | ressbas.b | . . . 4 ⊢ 𝐵 = (Base‘𝑊) | |
3 | 1, 2 | ressbas 16132 | . . 3 ⊢ (𝐴 ∈ V → (𝐴 ∩ 𝐵) = (Base‘𝑅)) |
4 | inss2 3977 | . . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵 | |
5 | 3, 4 | syl6eqssr 3797 | . 2 ⊢ (𝐴 ∈ V → (Base‘𝑅) ⊆ 𝐵) |
6 | reldmress 16128 | . . . . . 6 ⊢ Rel dom ↾s | |
7 | 6 | ovprc2 6848 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → (𝑊 ↾s 𝐴) = ∅) |
8 | 1, 7 | syl5eq 2806 | . . . 4 ⊢ (¬ 𝐴 ∈ V → 𝑅 = ∅) |
9 | 8 | fveq2d 6356 | . . 3 ⊢ (¬ 𝐴 ∈ V → (Base‘𝑅) = (Base‘∅)) |
10 | base0 16114 | . . . 4 ⊢ ∅ = (Base‘∅) | |
11 | 0ss 4115 | . . . 4 ⊢ ∅ ⊆ 𝐵 | |
12 | 10, 11 | eqsstr3i 3777 | . . 3 ⊢ (Base‘∅) ⊆ 𝐵 |
13 | 9, 12 | syl6eqss 3796 | . 2 ⊢ (¬ 𝐴 ∈ V → (Base‘𝑅) ⊆ 𝐵) |
14 | 5, 13 | pm2.61i 176 | 1 ⊢ (Base‘𝑅) ⊆ 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1632 ∈ wcel 2139 Vcvv 3340 ∩ cin 3714 ⊆ wss 3715 ∅c0 4058 ‘cfv 6049 (class class class)co 6813 Basecbs 16059 ↾s cress 16060 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-i2m1 10196 ax-1ne0 10197 ax-rrecex 10200 ax-cnre 10201 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-om 7231 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-nn 11213 df-ndx 16062 df-slot 16063 df-base 16065 df-sets 16066 df-ress 16067 |
This theorem is referenced by: funcres2c 16762 resscatc 16956 submnd0 17521 resscntz 17964 subcmn 18442 resspsrvsca 19620 subrgpsr 19621 ply1bascl 19775 evpmss 20134 frlmplusgval 20309 frlmvscafval 20311 lsslindf 20371 islinds3 20375 ressprdsds 22377 cphsubrglem 23177 |
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