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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 5958 | . 2 ⊢ Fun I | |
2 | resiexd.b | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
3 | resfunexg 6520 | . 2 ⊢ ((Fun I ∧ 𝐵 ∈ 𝑉) → ( I ↾ 𝐵) ∈ V) | |
4 | 1, 2, 3 | sylancr 696 | 1 ⊢ (𝜑 → ( I ↾ 𝐵) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2030 Vcvv 3231 I cid 5052 ↾ cres 5145 Fun wfun 5920 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pr 4936 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 |
This theorem is referenced by: estrcid 16821 funcestrcsetclem4 16830 funcestrcsetclem5 16831 funcsetcestrclem4 16845 funcsetcestrclem5 16846 cusgrsize 26406 rclexi 38239 cnvrcl0 38249 dfrtrcl5 38253 relexp01min 38322 uspgrsprfo 42081 funcrngcsetc 42323 funcrngcsetcALT 42324 funcringcsetc 42360 funcringcsetcALTV2lem4 42364 funcringcsetcALTV2lem5 42365 funcringcsetclem4ALTV 42387 funcringcsetclem5ALTV 42388 rhmsubcALTVlem3 42431 |
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