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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 6645 | . 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 19611 | . 2 ⊢ ∙ = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)) |
6 | fvex 6658 | . 2 ⊢ ( ∗ ‘𝐴) ∈ V | |
7 | 1, 5, 6 | fvmpt 6745 | 1 ⊢ (𝐴 ∈ 𝐵 → ( ∙ ‘𝐴) = ( ∗ ‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ‘cfv 6324 Basecbs 16475 *𝑟cstv 16559 *rfcstf 19607 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-staf 19609 |
This theorem is referenced by: srngcl 19619 srngnvl 19620 srngadd 19621 srngmul 19622 srng1 19623 srng0 19624 issrngd 19625 iporthcom 20324 |
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