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Mirrors > Home > MPE Home > Th. List > symgextfv | Structured version Visualization version GIF version |
Description: The function value of the extension of a permutation, fixing the additional element, for elements in the original domain. (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 |
---|---|
symgextfv | ⊢ ((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) → (𝑋 ∈ (𝑁 ∖ {𝐾}) → (𝐸‘𝑋) = (𝑍‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldifi 3732 | . . . 4 ⊢ (𝑋 ∈ (𝑁 ∖ {𝐾}) → 𝑋 ∈ 𝑁) | |
2 | fvexd 6203 | . . . . 5 ⊢ ((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) → (𝑍‘𝑋) ∈ V) | |
3 | ifexg 4157 | . . . . 5 ⊢ ((𝐾 ∈ 𝑁 ∧ (𝑍‘𝑋) ∈ V) → if(𝑋 = 𝐾, 𝐾, (𝑍‘𝑋)) ∈ V) | |
4 | 2, 3 | syldan 487 | . . . 4 ⊢ ((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) → if(𝑋 = 𝐾, 𝐾, (𝑍‘𝑋)) ∈ V) |
5 | eqeq1 2626 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 = 𝐾 ↔ 𝑋 = 𝐾)) | |
6 | fveq2 6191 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑍‘𝑥) = (𝑍‘𝑋)) | |
7 | 5, 6 | ifbieq2d 4111 | . . . . 5 ⊢ (𝑥 = 𝑋 → if(𝑥 = 𝐾, 𝐾, (𝑍‘𝑥)) = if(𝑋 = 𝐾, 𝐾, (𝑍‘𝑋))) |
8 | symgext.e | . . . . 5 ⊢ 𝐸 = (𝑥 ∈ 𝑁 ↦ if(𝑥 = 𝐾, 𝐾, (𝑍‘𝑥))) | |
9 | 7, 8 | fvmptg 6280 | . . . 4 ⊢ ((𝑋 ∈ 𝑁 ∧ if(𝑋 = 𝐾, 𝐾, (𝑍‘𝑋)) ∈ V) → (𝐸‘𝑋) = if(𝑋 = 𝐾, 𝐾, (𝑍‘𝑋))) |
10 | 1, 4, 9 | syl2anr 495 | . . 3 ⊢ (((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ∈ (𝑁 ∖ {𝐾})) → (𝐸‘𝑋) = if(𝑋 = 𝐾, 𝐾, (𝑍‘𝑋))) |
11 | eldifsn 4317 | . . . . . 6 ⊢ (𝑋 ∈ (𝑁 ∖ {𝐾}) ↔ (𝑋 ∈ 𝑁 ∧ 𝑋 ≠ 𝐾)) | |
12 | df-ne 2795 | . . . . . . . 8 ⊢ (𝑋 ≠ 𝐾 ↔ ¬ 𝑋 = 𝐾) | |
13 | 12 | biimpi 206 | . . . . . . 7 ⊢ (𝑋 ≠ 𝐾 → ¬ 𝑋 = 𝐾) |
14 | 13 | adantl 482 | . . . . . 6 ⊢ ((𝑋 ∈ 𝑁 ∧ 𝑋 ≠ 𝐾) → ¬ 𝑋 = 𝐾) |
15 | 11, 14 | sylbi 207 | . . . . 5 ⊢ (𝑋 ∈ (𝑁 ∖ {𝐾}) → ¬ 𝑋 = 𝐾) |
16 | 15 | adantl 482 | . . . 4 ⊢ (((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ∈ (𝑁 ∖ {𝐾})) → ¬ 𝑋 = 𝐾) |
17 | 16 | iffalsed 4097 | . . 3 ⊢ (((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ∈ (𝑁 ∖ {𝐾})) → if(𝑋 = 𝐾, 𝐾, (𝑍‘𝑋)) = (𝑍‘𝑋)) |
18 | 10, 17 | eqtrd 2656 | . 2 ⊢ (((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ∈ (𝑁 ∖ {𝐾})) → (𝐸‘𝑋) = (𝑍‘𝑋)) |
19 | 18 | ex 450 | 1 ⊢ ((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) → (𝑋 ∈ (𝑁 ∖ {𝐾}) → (𝐸‘𝑋) = (𝑍‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 Vcvv 3200 ∖ cdif 3571 ifcif 4086 {csn 4177 ↦ cmpt 4729 ‘cfv 5888 Basecbs 15857 SymGrpcsymg 17797 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 |
This theorem is referenced by: symgextf1lem 17840 symgextf1 17841 symgextfo 17842 symgextres 17845 |
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