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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 6598 | . 2 ⊢ Fun I | |
| 2 | ididg 5864 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 I 𝐴) | |
| 3 | funbrfv 6957 | . 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 1540 ∈ wcel 2108 class class class wbr 5143 I cid 5577 Fun wfun 6555 ‘cfv 6561 |
| 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-10 2141 ax-12 2177 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-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 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-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 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-iota 6514 df-fun 6563 df-fv 6569 |
| This theorem is referenced by: fviss 6986 fvmpti 7015 fvmpt2 7027 fvresi 7193 seqom0g 8496 fodomfi 9350 fodomfiOLD 9370 seqfeq4 14092 fac1 14316 facp1 14317 bcval5 14357 bcn2 14358 ids1 14635 s1val 14636 climshft2 15618 sum2id 15744 sumss 15760 prod2id 15964 fprodfac 16009 strfvi 17227 grpinvfvi 19000 mulgfvi 19091 efgrcl 19733 efgval 19735 frgp0 19778 frgpmhm 19783 vrgpf 19786 vrgpinv 19787 frgpupf 19791 frgpup1 19793 frgpup2 19794 frgpup3lem 19795 frgpnabllem1 19891 frgpnabllem2 19892 rlmsca2 21206 ply1basfvi 22242 ply1plusgfvi 22243 psr1sca2 22252 ply1sca2 22255 ply1scl0OLD 22294 ply1scl1OLD 22297 indislem 23007 2ndcctbss 23463 1stcelcls 23469 txindislem 23641 iscau3 25312 iscmet3 25327 ovolctb 25525 itg2splitlem 25783 deg1fvi 26124 deg1invg 26145 dgrle 26282 logfac 26643 fnpreimac 32681 ptpconn 35238 dicvscacl 41193 elinlem 43611 brfvid 43700 fvilbd 43702 tposid 48785 tposidres 48786 |
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