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Mirrors > Home > MPE Home > Th. List > fvi | Structured version Visualization version GIF version |
Description: The value of the identity function. (Contributed by NM, 1-May-2004.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
fvi | ⊢ (𝐴 ∈ 𝑉 → ( I ‘𝐴) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funi 5958 | . 2 ⊢ Fun I | |
2 | ididg 5308 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 I 𝐴) | |
3 | funbrfv 6272 | . 2 ⊢ (Fun I → (𝐴 I 𝐴 → ( I ‘𝐴) = 𝐴)) | |
4 | 1, 2, 3 | mpsyl 68 | 1 ⊢ (𝐴 ∈ 𝑉 → ( I ‘𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1523 ∈ wcel 2030 class class class wbr 4685 I cid 5052 Fun wfun 5920 ‘cfv 5926 |
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-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-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 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-br 4686 df-opab 4746 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-iota 5889 df-fun 5928 df-fv 5934 |
This theorem is referenced by: fviss 6295 fvmpti 6320 fvmpt2 6330 fvresi 6480 seqom0g 7596 fodomfi 8280 seqfeq4 12890 fac1 13104 facp1 13105 bcval5 13145 bcn2 13146 ids1 13413 s1val 13414 climshft2 14357 sum2id 14483 sumss 14499 prod2id 14702 fprodfac 14747 strfvi 15960 xpsc0 16267 xpsc1 16268 grpinvfvi 17510 mulgfvi 17592 efgrcl 18174 efgval 18176 frgp0 18219 frgpmhm 18224 vrgpf 18227 vrgpinv 18228 frgpupf 18232 frgpup1 18234 frgpup2 18235 frgpup3lem 18236 frgpnabllem1 18322 frgpnabllem2 18323 rlmsca2 19249 ply1basfvi 19659 ply1plusgfvi 19660 psr1sca2 19669 ply1sca2 19672 ply1scl0 19708 ply1scl1 19710 indislem 20852 2ndcctbss 21306 1stcelcls 21312 txindislem 21484 iscau3 23122 iscmet3 23137 ovolctb 23304 itg2splitlem 23560 deg1fvi 23890 deg1invg 23911 dgrle 24044 logfac 24392 ptpconn 31341 dicvscacl 36797 elinlem 38221 brfvid 38296 fvilbd 38298 |
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