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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 6411 | . 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 19165 | . 2 ⊢ ∙ = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)) |
6 | fvex 6424 | . 2 ⊢ ( ∗ ‘𝐴) ∈ V | |
7 | 1, 5, 6 | fvmpt 6507 | 1 ⊢ (𝐴 ∈ 𝐵 → ( ∙ ‘𝐴) = ( ∗ ‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1653 ∈ wcel 2157 ‘cfv 6101 Basecbs 16184 *𝑟cstv 16269 *rfcstf 19161 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-sbc 3634 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-br 4844 df-opab 4906 df-mpt 4923 df-id 5220 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-fv 6109 df-staf 19163 |
This theorem is referenced by: srngcl 19173 srngnvl 19174 srngadd 19175 srngmul 19176 srng1 19177 srng0 19178 issrngd 19179 iporthcom 20304 |
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