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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 6010 | . 2 ⊢ (℩𝑒(𝑒 ∈ (Base‘𝑔) ∧ ∀𝑥 ∈ (Base‘𝑔)((𝑒(+g‘𝑔)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑔)𝑒) = 𝑥))) ∈ V | |
2 | df-0g 16310 | . 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 6161 | 1 ⊢ 0g Fn V |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 382 = wceq 1631 ∈ wcel 2145 ∀wral 3061 Vcvv 3351 ℩cio 5991 Fn wfn 6025 ‘cfv 6030 (class class class)co 6796 Basecbs 16064 +gcplusg 16149 0gc0g 16308 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pr 5035 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4227 df-sn 4318 df-pr 4320 df-op 4324 df-uni 4576 df-br 4788 df-opab 4848 df-mpt 4865 df-id 5158 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-iota 5993 df-fun 6032 df-fn 6033 df-0g 16310 |
This theorem is referenced by: prdsidlem 17530 pws0g 17534 prdsinvlem 17732 pws1 18824 dsmmbas2 20298 frlmbas 20316 |
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