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| Mirrors > Home > MPE Home > Th. List > reseq2i | Structured version Visualization version GIF version | ||
| Description: Equality inference for restrictions. (Contributed by Paul Chapman, 22-Jun-2011.) |
| Ref | Expression |
|---|---|
| reseqi.1 | ⊢ 𝐴 = 𝐵 |
| Ref | Expression |
|---|---|
| reseq2i | ⊢ (𝐶 ↾ 𝐴) = (𝐶 ↾ 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | reseqi.1 | . 2 ⊢ 𝐴 = 𝐵 | |
| 2 | reseq2 5992 | . 2 ⊢ (𝐴 = 𝐵 → (𝐶 ↾ 𝐴) = (𝐶 ↾ 𝐵)) | |
| 3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐶 ↾ 𝐴) = (𝐶 ↾ 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ↾ cres 5687 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-in 3958 df-opab 5206 df-xp 5691 df-res 5697 |
| This theorem is referenced by: reseq12i 5995 rescom 6020 resdmdfsn 6049 idinxpresid 6066 rescnvcnv 6224 resdm2 6251 funcnvres 6644 resasplit 6778 fresaunres2 6780 fresaunres1 6781 resdif 6869 resin 6870 funcocnv2 6873 fvn0ssdmfun 7094 residpr 7163 eqfunressuc 7381 fprlem1 8325 wfrlem5OLD 8353 domss2 9176 ordtypelem1 9558 frrlem15 9797 ackbij2lem3 10280 facnn 14314 fac0 14315 hashresfn 14379 relexpcnv 15074 divcnvshft 15891 ruclem4 16270 fsets 17206 setsid 17244 join0 18450 meet0 18451 symgfixelsi 19453 psgnsn 19538 dprd2da 20062 ply1plusgfvi 22243 uptx 23633 txcn 23634 ressxms 24538 ressms 24539 iscmet3lem3 25324 volres 25563 dvlip 26032 dvne0 26050 lhop 26055 dflog2 26602 dfrelog 26607 dvlog 26693 wilthlem2 27112 nosupbnd2lem1 27760 noinfbnd2lem1 27775 0grsubgr 29295 0pth 30144 1pthdlem1 30154 eupth2lemb 30256 ex-fpar 30481 fressupp 32697 df1stres 32713 df2ndres 32714 ffsrn 32740 resf1o 32741 fpwrelmapffs 32745 cycpmconjv 33162 sitmcl 34353 eulerpartlemn 34383 bnj1326 35040 satfv1lem 35367 divcnvlin 35733 poimirlem9 37636 zrdivrng 37960 isdrngo1 37963 ressucdifsn 38246 cnvresrn 38349 disjsuc 38760 eldioph4b 42822 diophren 42824 rclexi 43628 rtrclex 43630 cnvrcl0 43638 dfrtrcl5 43642 dfrcl2 43687 relexpiidm 43717 relexp01min 43726 relexpaddss 43731 seff 44328 sblpnf 44329 radcnvrat 44333 hashnzfzclim 44341 dvresioo 45936 fourierdlem72 46193 fourierdlem80 46201 fourierdlem94 46215 fourierdlem103 46224 fourierdlem104 46225 fourierdlem113 46234 fouriersw 46246 sge0split 46424 isubgrgrim 47897 stgr0 47927 stgr1 47928 rngcidALTV 48190 ringcidALTV 48224 tposresg 48778 tposres3 48781 tposresxp 48783 |
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