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Mirrors > Home > MPE Home > Th. List > fn0g | Structured version Visualization version GIF version |
Description: The group zero extractor is a function. (Contributed by Stefan O'Rear, 10-Jan-2015.) |
Ref | Expression |
---|---|
fn0g | ⊢ 0g Fn V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iotaex 6413 | . 2 ⊢ (℩𝑒(𝑒 ∈ (Base‘𝑔) ∧ ∀𝑥 ∈ (Base‘𝑔)((𝑒(+g‘𝑔)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑔)𝑒) = 𝑥))) ∈ V | |
2 | df-0g 17152 | . 2 ⊢ 0g = (𝑔 ∈ V ↦ (℩𝑒(𝑒 ∈ (Base‘𝑔) ∧ ∀𝑥 ∈ (Base‘𝑔)((𝑒(+g‘𝑔)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑔)𝑒) = 𝑥)))) | |
3 | 1, 2 | fnmpti 6576 | 1 ⊢ 0g Fn V |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∀wral 3064 Vcvv 3432 ℩cio 6389 Fn wfn 6428 ‘cfv 6433 (class class class)co 7275 Basecbs 16912 +gcplusg 16962 0gc0g 17150 |
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-pr 5352 |
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-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-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-iota 6391 df-fun 6435 df-fn 6436 df-0g 17152 |
This theorem is referenced by: prdsidlem 18417 pws0g 18421 prdsinvlem 18684 pws1 19855 dsmmbas2 20944 frlmbas 20962 |
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