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| Mirrors > Home > MPE Home > Th. List > fnresi | Structured version Visualization version GIF version | ||
| Description: The restricted identity relation is a function on the restricting class. (Contributed by NM, 27-Aug-2004.) (Proof shortened by BJ, 27-Dec-2023.) | 
| Ref | Expression | 
|---|---|
| fnresi | ⊢ ( I ↾ 𝐴) Fn 𝐴 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | idfn 6696 | . 2 ⊢ I Fn V | |
| 2 | ssv 4008 | . 2 ⊢ 𝐴 ⊆ V | |
| 3 | fnssres 6691 | . 2 ⊢ (( I Fn V ∧ 𝐴 ⊆ V) → ( I ↾ 𝐴) Fn 𝐴) | |
| 4 | 1, 2, 3 | mp2an 692 | 1 ⊢ ( I ↾ 𝐴) Fn 𝐴 | 
| Colors of variables: wff setvar class | 
| Syntax hints: Vcvv 3480 ⊆ wss 3951 I cid 5577 ↾ cres 5687 Fn wfn 6556 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-res 5697 df-fun 6563 df-fn 6564 | 
| This theorem is referenced by: f1oi 6886 fninfp 7194 fndifnfp 7196 fnnfpeq0 7198 fveqf1o 7322 weniso 7374 iordsmo 8397 fipreima 9398 dfac9 10177 smndex1n0mnd 18925 pmtrfinv 19479 psdmplcl 22166 ustuqtop3 24252 fta1blem 26210 qaa 26365 dfiop2 31772 symgcom2 33104 tocycfvres1 33130 tocycfvres2 33131 cvmliftlem4 35293 cvmliftlem5 35294 poimirlem15 37642 poimirlem22 37649 ltrnid 40137 dvsid 44350 dflinc2 48327 tposideq 48788 | 
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