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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 6586 | . 2 ⊢ Fun I | |
2 | ididg 5856 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 I 𝐴) | |
3 | funbrfv 6947 | . 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 1533 ∈ wcel 2098 class class class wbr 5149 I cid 5575 Fun wfun 6543 ‘cfv 6549 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-iota 6501 df-fun 6551 df-fv 6557 |
This theorem is referenced by: fviss 6974 fvmpti 7003 fvmpt2 7015 fvresi 7182 seqom0g 8477 fodomfi 9356 seqfeq4 14057 fac1 14280 facp1 14281 bcval5 14321 bcn2 14322 ids1 14591 s1val 14592 climshft2 15570 sum2id 15698 sumss 15714 prod2id 15916 fprodfac 15961 strfvi 17178 grpinvfvi 18963 mulgfvi 19053 efgrcl 19699 efgval 19701 frgp0 19744 frgpmhm 19749 vrgpf 19752 vrgpinv 19753 frgpupf 19757 frgpup1 19759 frgpup2 19760 frgpup3lem 19761 frgpnabllem1 19857 frgpnabllem2 19858 rlmsca2 21121 ply1basfvi 22200 ply1plusgfvi 22201 psr1sca2 22210 ply1sca2 22213 ply1scl0OLD 22252 ply1scl1OLD 22255 indislem 22964 2ndcctbss 23420 1stcelcls 23426 txindislem 23598 iscau3 25267 iscmet3 25282 ovolctb 25480 itg2splitlem 25739 deg1fvi 26082 deg1invg 26103 dgrle 26239 logfac 26597 fnpreimac 32558 ptpconn 34994 dicvscacl 40814 elinlem 43175 brfvid 43264 fvilbd 43266 |
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