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Mirrors > Home > MPE Home > Th. List > fvresi | Structured version Visualization version GIF version |
Description: The value of a restricted identity function. (Contributed by NM, 19-May-2004.) |
Ref | Expression |
---|---|
fvresi | ⊢ (𝐵 ∈ 𝐴 → (( I ↾ 𝐴)‘𝐵) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvres 6117 | . 2 ⊢ (𝐵 ∈ 𝐴 → (( I ↾ 𝐴)‘𝐵) = ( I ‘𝐵)) | |
2 | fvi 6165 | . 2 ⊢ (𝐵 ∈ 𝐴 → ( I ‘𝐵) = 𝐵) | |
3 | 1, 2 | eqtrd 2644 | 1 ⊢ (𝐵 ∈ 𝐴 → (( I ↾ 𝐴)‘𝐵) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 I cid 4948 ↾ cres 5040 ‘cfv 5804 |
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-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-res 5050 df-iota 5768 df-fun 5806 df-fv 5812 |
This theorem is referenced by: fninfp 6345 fndifnfp 6347 fnnfpeq0 6349 f1ocnvfv1 6432 f1ocnvfv2 6433 fcof1 6442 fcofo 6443 isoid 6479 weniso 6504 iordsmo 7341 fipreima 8155 infxpenc 8724 dfac9 8841 fproddvdsd 14897 ndxarg 15715 idfu2 16361 idfu1 16363 idfucl 16364 cofurid 16374 funcestrcsetclem6 16608 funcestrcsetclem7 16609 funcestrcsetclem9 16611 funcsetcestrclem6 16623 funcsetcestrclem7 16624 funcsetcestrclem9 16626 yonedainv 16744 idmhm 17167 idghm 17498 lactghmga 17647 symgga 17649 cayleylem2 17656 gsmsymgrfix 17671 gsmsymgreq 17675 pmtrfinv 17704 idlmhm 18862 evl1vard 19522 islinds2 19971 lindsind2 19977 madetsumid 20086 mdetunilem7 20243 txkgen 21265 ustuqtop3 21857 iducn 21897 nmoid 22356 dvid 23487 mvth 23559 fta1blem 23732 qaa 23882 idmot 25232 dfiop2 27996 idunop 28221 idcnop 28224 elunop2 28256 lnophm 28262 qqhre 29392 subfacp1lem4 30419 subfacp1lem5 30420 cvmliftlem5 30525 idlaut 34400 idldil 34418 ltrnid 34439 idltrn 34454 ltrnideq 34480 tendoidcl 35075 tendo1ne0 35134 cdleml7 35288 tendospid 35324 dvalveclem 35332 rngunsnply 36762 idmgmhm 41578 funcrngcsetcALT 41791 funcringcsetcALTV2lem6 41833 funcringcsetcALTV2lem7 41834 funcringcsetcALTV2lem9 41836 funcringcsetclem6ALTV 41856 funcringcsetclem7ALTV 41857 funcringcsetclem9ALTV 41859 dflinc2 41993 |
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