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| Mirrors > Home > MPE Home > Th. List > stafval | Structured version Visualization version GIF version | ||
| Description: The functionalization of the involution component of a structure. (Contributed by Mario Carneiro, 6-Oct-2015.) | 
| Ref | Expression | 
|---|---|
| staffval.b | ⊢ 𝐵 = (Base‘𝑅) | 
| staffval.i | ⊢ ∗ = (*𝑟‘𝑅) | 
| staffval.f | ⊢ ∙ = (*rf‘𝑅) | 
| Ref | Expression | 
|---|---|
| stafval | ⊢ (𝐴 ∈ 𝐵 → ( ∙ ‘𝐴) = ( ∗ ‘𝐴)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | fveq2 6905 | . 2 ⊢ (𝑥 = 𝐴 → ( ∗ ‘𝑥) = ( ∗ ‘𝐴)) | |
| 2 | staffval.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
| 3 | staffval.i | . . 3 ⊢ ∗ = (*𝑟‘𝑅) | |
| 4 | staffval.f | . . 3 ⊢ ∙ = (*rf‘𝑅) | |
| 5 | 2, 3, 4 | staffval 20843 | . 2 ⊢ ∙ = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)) | 
| 6 | fvex 6918 | . 2 ⊢ ( ∗ ‘𝐴) ∈ V | |
| 7 | 1, 5, 6 | fvmpt 7015 | 1 ⊢ (𝐴 ∈ 𝐵 → ( ∙ ‘𝐴) = ( ∗ ‘𝐴)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 ‘cfv 6560 Basecbs 17248 *𝑟cstv 17300 *rfcstf 20839 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-fv 6568 df-staf 20841 | 
| This theorem is referenced by: srngcl 20851 srngnvl 20852 srngadd 20853 srngmul 20854 srng1 20855 srng0 20856 issrngd 20857 iporthcom 21654 | 
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