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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 | dffn5 6717 | . . 3 ⊢ ( ∗ Fn 𝐵 ↔ ∗ = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥))) | |
2 | 1 | biimpi 217 | . 2 ⊢ ( ∗ Fn 𝐵 → ∗ = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥))) |
3 | staffval.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
4 | staffval.i | . . 3 ⊢ ∗ = (*𝑟‘𝑅) | |
5 | staffval.f | . . 3 ⊢ ∙ = (*rf‘𝑅) | |
6 | 3, 4, 5 | staffval 19547 | . 2 ⊢ ∙ = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)) |
7 | 2, 6 | syl6reqr 2872 | 1 ⊢ ( ∗ Fn 𝐵 → ∙ = ∗ ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ↦ cmpt 5137 Fn wfn 6343 ‘cfv 6348 Basecbs 16471 *𝑟cstv 16555 *rfcstf 19543 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 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 19545 |
This theorem is referenced by: (None) |
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