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Mirrors > Home > MPE Home > Th. List > resiexd | Structured version Visualization version GIF version |
Description: The restriction of the identity relation to a set is a set. (Contributed by AV, 15-Feb-2020.) |
Ref | Expression |
---|---|
resiexd.b | ⊢ (𝜑 → 𝐵 ∈ 𝑉) |
Ref | Expression |
---|---|
resiexd | ⊢ (𝜑 → ( I ↾ 𝐵) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funi 6380 | . 2 ⊢ Fun I | |
2 | resiexd.b | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
3 | resfunexg 6969 | . 2 ⊢ ((Fun I ∧ 𝐵 ∈ 𝑉) → ( I ↾ 𝐵) ∈ V) | |
4 | 1, 2, 3 | sylancr 587 | 1 ⊢ (𝜑 → ( I ↾ 𝐵) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 Vcvv 3492 I cid 5452 ↾ cres 5550 Fun wfun 6342 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 |
This theorem is referenced by: setcid 17334 estrcid 17372 funcestrcsetclem5 17382 funcsetcestrclem5 17397 cusgrsize 27163 tocycfv 30678 tocycf 30686 lindspropd 30870 rclexi 39853 cnvrcl0 39863 dfrtrcl5 39867 relexp01min 39936 fundcmpsurbijinjpreimafv 43444 fundcmpsurinjALT 43449 isomgreqve 43867 ushrisomgr 43883 uspgrsprfo 43900 funcrngcsetc 44197 funcrngcsetcALT 44198 funcringcsetc 44234 funcringcsetcALTV2lem4 44238 funcringcsetcALTV2lem5 44239 funcringcsetclem4ALTV 44261 funcringcsetclem5ALTV 44262 rhmsubcALTVlem3 44305 |
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