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Mirrors > Home > MPE Home > Th. List > symgfixels | Structured version Visualization version GIF version |
Description: The restriction of a permutation to a set with one element removed is an element of the restricted symmetric group if the restriction is a 1-1 onto function. (Contributed by AV, 4-Jan-2019.) |
Ref | Expression |
---|---|
symgfixf.p | ⊢ 𝑃 = (Base‘(SymGrp‘𝑁)) |
symgfixf.q | ⊢ 𝑄 = {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾} |
symgfixf.s | ⊢ 𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾}))) |
symgfixf.d | ⊢ 𝐷 = (𝑁 ∖ {𝐾}) |
Ref | Expression |
---|---|
symgfixels | ⊢ (𝐹 ∈ 𝑉 → ((𝐹 ↾ 𝐷) ∈ 𝑆 ↔ (𝐹 ↾ 𝐷):𝐷–1-1-onto→𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | symgfixf.s | . . . 4 ⊢ 𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾}))) | |
2 | 1 | eleq2i 2819 | . . 3 ⊢ ((𝐹 ↾ 𝐷) ∈ 𝑆 ↔ (𝐹 ↾ 𝐷) ∈ (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))) |
3 | 2 | a1i 11 | . 2 ⊢ (𝐹 ∈ 𝑉 → ((𝐹 ↾ 𝐷) ∈ 𝑆 ↔ (𝐹 ↾ 𝐷) ∈ (Base‘(SymGrp‘(𝑁 ∖ {𝐾}))))) |
4 | resexg 6020 | . . 3 ⊢ (𝐹 ∈ 𝑉 → (𝐹 ↾ 𝐷) ∈ V) | |
5 | eqid 2726 | . . . 4 ⊢ (SymGrp‘(𝑁 ∖ {𝐾})) = (SymGrp‘(𝑁 ∖ {𝐾})) | |
6 | eqid 2726 | . . . 4 ⊢ (Base‘(SymGrp‘(𝑁 ∖ {𝐾}))) = (Base‘(SymGrp‘(𝑁 ∖ {𝐾}))) | |
7 | 5, 6 | elsymgbas2 19289 | . . 3 ⊢ ((𝐹 ↾ 𝐷) ∈ V → ((𝐹 ↾ 𝐷) ∈ (Base‘(SymGrp‘(𝑁 ∖ {𝐾}))) ↔ (𝐹 ↾ 𝐷):(𝑁 ∖ {𝐾})–1-1-onto→(𝑁 ∖ {𝐾}))) |
8 | 4, 7 | syl 17 | . 2 ⊢ (𝐹 ∈ 𝑉 → ((𝐹 ↾ 𝐷) ∈ (Base‘(SymGrp‘(𝑁 ∖ {𝐾}))) ↔ (𝐹 ↾ 𝐷):(𝑁 ∖ {𝐾})–1-1-onto→(𝑁 ∖ {𝐾}))) |
9 | eqidd 2727 | . . 3 ⊢ (𝐹 ∈ 𝑉 → (𝐹 ↾ 𝐷) = (𝐹 ↾ 𝐷)) | |
10 | symgfixf.d | . . . . 5 ⊢ 𝐷 = (𝑁 ∖ {𝐾}) | |
11 | 10 | a1i 11 | . . . 4 ⊢ (𝐹 ∈ 𝑉 → 𝐷 = (𝑁 ∖ {𝐾})) |
12 | 11 | eqcomd 2732 | . . 3 ⊢ (𝐹 ∈ 𝑉 → (𝑁 ∖ {𝐾}) = 𝐷) |
13 | 9, 12, 12 | f1oeq123d 6820 | . 2 ⊢ (𝐹 ∈ 𝑉 → ((𝐹 ↾ 𝐷):(𝑁 ∖ {𝐾})–1-1-onto→(𝑁 ∖ {𝐾}) ↔ (𝐹 ↾ 𝐷):𝐷–1-1-onto→𝐷)) |
14 | 3, 8, 13 | 3bitrd 305 | 1 ⊢ (𝐹 ∈ 𝑉 → ((𝐹 ↾ 𝐷) ∈ 𝑆 ↔ (𝐹 ↾ 𝐷):𝐷–1-1-onto→𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 ∈ wcel 2098 {crab 3426 Vcvv 3468 ∖ cdif 3940 {csn 4623 ↾ cres 5671 –1-1-onto→wf1o 6535 ‘cfv 6536 Basecbs 17150 SymGrpcsymg 19283 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-uz 12824 df-fz 13488 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17151 df-ress 17180 df-plusg 17216 df-tset 17222 df-efmnd 18791 df-symg 19284 |
This theorem is referenced by: symgfixelsi 19352 |
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