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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 5834 | . 2 ⊢ Fun I | |
2 | resiexd.b | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
3 | resfunexg 6384 | . 2 ⊢ ((Fun I ∧ 𝐵 ∈ 𝑉) → ( I ↾ 𝐵) ∈ V) | |
4 | 1, 2, 3 | sylancr 694 | 1 ⊢ (𝜑 → ( I ↾ 𝐵) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 Vcvv 3173 I cid 4948 ↾ cres 5040 Fun wfun 5798 |
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-rep 4699 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-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 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-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 |
This theorem is referenced by: estrcid 16597 funcestrcsetclem4 16606 funcestrcsetclem5 16607 funcsetcestrclem4 16621 funcsetcestrclem5 16622 rclexi 36941 cnvrcl0 36951 dfrtrcl5 36955 relexp01min 37024 cusgrsize 40670 funcrngcsetc 41790 funcrngcsetcALT 41791 funcringcsetc 41827 funcringcsetcALTV2lem4 41831 funcringcsetcALTV2lem5 41832 funcringcsetclem4ALTV 41854 funcringcsetclem5ALTV 41855 rhmsubcALTVlem3 41899 |
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