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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 5312 | . 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 1475 ↾ 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 |
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-opab 4644 df-xp 5044 df-res 5050 |
This theorem is referenced by: reseq12i 5315 rescom 5343 resdmdfsn 5365 rescnvcnv 5515 resdm2 5542 funcnvres 5881 resasplit 5987 fresaunres2 5989 fresaunres1 5990 resdif 6070 resin 6071 funcocnv2 6074 fvn0ssdmfun 6258 residpr 6315 wfrlem5 7306 domss2 8004 ordtypelem1 8306 ackbij2lem3 8946 facnn 12924 fac0 12925 hashresfn 12990 relexpcnv 13623 divcnvshft 14426 ruclem4 14802 fsets 15723 setsid 15742 meet0 16960 join0 16961 symgfixelsi 17678 psgnsn 17763 dprd2da 18264 ply1plusgfvi 19433 uptx 21238 txcn 21239 ressxms 22140 ressms 22141 iscmet3lem3 22896 volres 23103 dvlip 23560 dvne0 23578 lhop 23583 dflog2 24111 dfrelog 24116 dvlog 24197 wilthlem2 24595 0pth 26100 2pthlem1 26125 df1stres 28864 df2ndres 28865 ffsrn 28892 resf1o 28893 fpwrelmapffs 28897 sitmcl 29740 eulerpartlemn 29770 bnj1326 30348 divcnvlin 30871 frrlem5 31028 poimirlem9 32588 zrdivrng 32922 isdrngo1 32925 eldioph4b 36393 diophren 36395 rclexi 36941 rtrclex 36943 cnvrcl0 36951 dfrtrcl5 36955 dfrcl2 36985 relexpiidm 37015 relexp01min 37024 relexpaddss 37029 seff 37530 sblpnf 37531 radcnvrat 37535 hashnzfzclim 37543 dvresioo 38811 fourierdlem72 39071 fourierdlem80 39079 fourierdlem94 39093 fourierdlem103 39102 fourierdlem104 39103 fourierdlem113 39112 fouriersw 39124 sge0split 39302 0grsubgr 40502 0pth-av 41293 1pthdlem1 41302 eupth2lemb 41405 rngcidALTV 41783 ringcidALTV 41846 |
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