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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 6663 | . 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 19610 | . 2 ⊢ ∙ = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)) |
6 | fvex 6676 | . 2 ⊢ ( ∗ ‘𝐴) ∈ V | |
7 | 1, 5, 6 | fvmpt 6761 | 1 ⊢ (𝐴 ∈ 𝐵 → ( ∙ ‘𝐴) = ( ∗ ‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1531 ∈ wcel 2108 ‘cfv 6348 Basecbs 16475 *𝑟cstv 16559 *rfcstf 19606 |
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 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-sbc 3771 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-staf 19608 |
This theorem is referenced by: srngcl 19618 srngnvl 19619 srngadd 19620 srngmul 19621 srng1 19622 srng0 19623 issrngd 19624 iporthcom 20771 |
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