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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 20107 | . 2 ⊢ ∙ = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥)) |
5 | dffn5 6828 | . . 3 ⊢ ( ∗ Fn 𝐵 ↔ ∗ = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥))) | |
6 | 5 | biimpi 215 | . 2 ⊢ ( ∗ Fn 𝐵 → ∗ = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥))) |
7 | 4, 6 | eqtr4id 2797 | 1 ⊢ ( ∗ Fn 𝐵 → ∙ = ∗ ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ↦ cmpt 5157 Fn wfn 6428 ‘cfv 6433 Basecbs 16912 *𝑟cstv 16964 *rfcstf 20103 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-fv 6441 df-staf 20105 |
This theorem is referenced by: (None) |
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