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Mirrors > Home > MPE Home > Th. List > resex | Structured version Visualization version GIF version |
Description: The restriction of a set is a set. (Contributed by Jeff Madsen, 19-Jun-2011.) |
Ref | Expression |
---|---|
resex.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
resex | ⊢ (𝐴 ↾ 𝐵) ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resex.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | resexg 5362 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ↾ 𝐵) ∈ V) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐴 ↾ 𝐵) ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1977 Vcvv 3173 ↾ cres 5040 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-in 3547 df-ss 3554 df-res 5050 |
This theorem is referenced by: wfrlem17 7318 tfrlem9a 7369 domunsncan 7945 sbthlem10 7964 mapunen 8014 php3 8031 ssfi 8065 marypha1lem 8222 infdifsn 8437 ackbij2lem3 8946 fin1a2lem7 9111 hashf1lem2 13097 ramub2 15556 resf1st 16377 resf2nd 16378 funcres 16379 lubfval 16801 glbfval 16814 znval 19702 znle 19703 usgrafis 25944 cusgrasize 26006 wlknwwlknvbij 26268 clwwlkvbij 26329 hhssva 27498 hhsssm 27499 hhssnm 27500 hhshsslem1 27508 eulerpartlemt 29760 eulerpartgbij 29761 eulerpart 29771 fibp1 29790 subfacp1lem3 30418 subfacp1lem5 30420 dfrdg2 30945 nofulllem5 31105 dfrecs2 31227 finixpnum 32564 poimirlem4 32583 poimirlem9 32588 mbfresfi 32626 sdclem2 32708 diophrex 36357 rexrabdioph 36376 2rexfrabdioph 36378 3rexfrabdioph 36379 4rexfrabdioph 36380 6rexfrabdioph 36381 7rexfrabdioph 36382 rmydioph 36599 rmxdioph 36601 expdiophlem2 36607 ssnnf1octb 38377 dvnprodlem1 38836 dvnprodlem2 38837 fouriersw 39124 vonval 39430 hoidmvlelem2 39486 hoidmvlelem3 39487 iccelpart 39971 uhgrspanop 40520 upgrspanop 40521 umgrspanop 40522 usgrspanop 40523 uhgrspan1lem1 40524 wlksnwwlknvbij 41114 clwwlksvbij 41229 eupthvdres 41403 eupth2lem3 41404 eupth2lemb 41405 |
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