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Mirrors > Home > MPE Home > Th. List > symgextfve | Structured version Visualization version GIF version |
Description: The function value of the extension of a permutation, fixing the additional element, for the additional element. (Contributed by AV, 6-Jan-2019.) |
Ref | Expression |
---|---|
symgext.s | ⊢ 𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾}))) |
symgext.e | ⊢ 𝐸 = (𝑥 ∈ 𝑁 ↦ if(𝑥 = 𝐾, 𝐾, (𝑍‘𝑥))) |
Ref | Expression |
---|---|
symgextfve | ⊢ (𝐾 ∈ 𝑁 → (𝑋 = 𝐾 → (𝐸‘𝑋) = 𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6896 | . . 3 ⊢ (𝑋 = 𝐾 → (𝐸‘𝑋) = (𝐸‘𝐾)) | |
2 | iftrue 4536 | . . . . 5 ⊢ (𝑥 = 𝐾 → if(𝑥 = 𝐾, 𝐾, (𝑍‘𝑥)) = 𝐾) | |
3 | symgext.e | . . . . 5 ⊢ 𝐸 = (𝑥 ∈ 𝑁 ↦ if(𝑥 = 𝐾, 𝐾, (𝑍‘𝑥))) | |
4 | 2, 3 | fvmptg 7002 | . . . 4 ⊢ ((𝐾 ∈ 𝑁 ∧ 𝐾 ∈ 𝑁) → (𝐸‘𝐾) = 𝐾) |
5 | 4 | anidms 565 | . . 3 ⊢ (𝐾 ∈ 𝑁 → (𝐸‘𝐾) = 𝐾) |
6 | 1, 5 | sylan9eqr 2787 | . 2 ⊢ ((𝐾 ∈ 𝑁 ∧ 𝑋 = 𝐾) → (𝐸‘𝑋) = 𝐾) |
7 | 6 | ex 411 | 1 ⊢ (𝐾 ∈ 𝑁 → (𝑋 = 𝐾 → (𝐸‘𝑋) = 𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∖ cdif 3941 ifcif 4530 {csn 4630 ↦ cmpt 5232 ‘cfv 6549 Basecbs 17183 SymGrpcsymg 19333 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-iota 6501 df-fun 6551 df-fv 6557 |
This theorem is referenced by: symgextf1lem 19387 symgextfo 19389 |
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