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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.) (Proof shortened by SN, 25-Feb-2025.) |
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 | . . 3 ⊢ 𝑅 = (𝑊 ↾s 𝐴) | |
2 | ressbas.b | . . 3 ⊢ 𝐵 = (Base‘𝑊) | |
3 | 1, 2 | ressbasssg 17216 | . 2 ⊢ (Base‘𝑅) ⊆ (𝐴 ∩ 𝐵) |
4 | inss2 4224 | . 2 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵 | |
5 | 3, 4 | sstri 3982 | 1 ⊢ (Base‘𝑅) ⊆ 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∩ cin 3938 ⊆ wss 3939 ‘cfv 6543 (class class class)co 7416 Basecbs 17179 ↾s cress 17208 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-1cn 11196 ax-addcl 11198 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-nn 12243 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 |
This theorem is referenced by: funcres2c 17889 resscatc 18097 submnd0 18722 resscntz 19288 subcmn 19796 rngqiprng1elbas 21180 rng2idl1cntr 21199 evpmss 21522 phlssphl 21595 frlmplusgval 21702 frlmvscafval 21704 lsslindf 21768 islinds3 21772 resspsrvsca 21926 subrgpsr 21927 ply1bascl 22131 evls1fvcl 22303 ressprdsds 24295 cphsubrglem 25123 cphsscph 25197 ressply1mon1p 33310 mplsubrgcl 41836 |
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