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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 6600 | . 2 ⊢ Fun I | |
2 | resiexd.b | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
3 | resfunexg 7235 | . 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 2106 Vcvv 3478 I cid 5582 ↾ cres 5691 Fun wfun 6557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 |
This theorem is referenced by: setcid 18140 estrcid 18189 funcestrcsetclem5 18200 funcsetcestrclem5 18215 funcrngcsetc 20657 funcrngcsetcALT 20658 funcringcsetc 20691 cusgrsize 29487 fzo0pmtrlast 33095 tocycfv 33112 tocycf 33120 lindspropd 33391 rclexi 43605 cnvrcl0 43615 dfrtrcl5 43619 relexp01min 43703 fundcmpsurbijinjpreimafv 47332 fundcmpsurinjALT 47337 ushggricedg 47834 stgrvtx 47857 stgriedg 47858 grlicref 47908 gpgvtx 47938 gpgiedg 47939 uspgrsprfo 47992 rhmsubcALTVlem3 48127 funcringcsetcALTV2lem4 48137 funcringcsetcALTV2lem5 48138 funcringcsetclem4ALTV 48160 funcringcsetclem5ALTV 48161 itcoval0 48512 itcoval1 48513 |
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