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Mirrors > Home > MPE Home > Th. List > staffn | Structured version Visualization version GIF version |
Description: The functionalization is equal to the original function, if it is a function on the right base set. (Contributed by Mario Carneiro, 6-Oct-2015.) |
Ref | Expression |
---|---|
staffval.b | ⊢ 𝐵 = (Base‘𝑅) |
staffval.i | ⊢ ∗ = (*𝑟‘𝑅) |
staffval.f | ⊢ ∙ = (*rf‘𝑅) |
Ref | Expression |
---|---|
staffn | ⊢ ( ∗ Fn 𝐵 → ∙ = ∗ ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | staffval.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
2 | staffval.i | . . 3 ⊢ ∗ = (*𝑟‘𝑅) | |
3 | staffval.f | . . 3 ⊢ ∙ = (*rf‘𝑅) | |
4 | 1, 2, 3 | staffval 19737 | . 2 ⊢ ∙ = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)) |
5 | dffn5 6728 | . . 3 ⊢ ( ∗ Fn 𝐵 ↔ ∗ = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥))) | |
6 | 5 | biimpi 219 | . 2 ⊢ ( ∗ Fn 𝐵 → ∗ = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥))) |
7 | 4, 6 | eqtr4id 2792 | 1 ⊢ ( ∗ Fn 𝐵 → ∙ = ∗ ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ↦ cmpt 5110 Fn wfn 6334 ‘cfv 6339 Basecbs 16586 *𝑟cstv 16670 *rfcstf 19733 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-sbc 3681 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-fv 6347 df-staf 19735 |
This theorem is referenced by: (None) |
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