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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 6901 | . 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 20820 | . 2 ⊢ ∙ = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)) |
6 | fvex 6914 | . 2 ⊢ ( ∗ ‘𝐴) ∈ V | |
7 | 1, 5, 6 | fvmpt 7009 | 1 ⊢ (𝐴 ∈ 𝐵 → ( ∙ ‘𝐴) = ( ∗ ‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ‘cfv 6554 Basecbs 17213 *𝑟cstv 17268 *rfcstf 20816 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-fv 6562 df-staf 20818 |
This theorem is referenced by: srngcl 20828 srngnvl 20829 srngadd 20830 srngmul 20831 srng1 20832 srng0 20833 issrngd 20834 iporthcom 21631 |
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