Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2906 | . . 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 5900 | . . 3 ⊢ (𝐹 ∈ 𝑉 → (𝐹 ↾ 𝐷) ∈ V) | |
5 | eqid 2823 | . . . 4 ⊢ (SymGrp‘(𝑁 ∖ {𝐾})) = (SymGrp‘(𝑁 ∖ {𝐾})) | |
6 | eqid 2823 | . . . 4 ⊢ (Base‘(SymGrp‘(𝑁 ∖ {𝐾}))) = (Base‘(SymGrp‘(𝑁 ∖ {𝐾}))) | |
7 | 5, 6 | elsymgbas2 18503 | . . 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 2824 | . . 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 2829 | . . 3 ⊢ (𝐹 ∈ 𝑉 → (𝑁 ∖ {𝐾}) = 𝐷) |
13 | 9, 12, 12 | f1oeq123d 6612 | . 2 ⊢ (𝐹 ∈ 𝑉 → ((𝐹 ↾ 𝐷):(𝑁 ∖ {𝐾})–1-1-onto→(𝑁 ∖ {𝐾}) ↔ (𝐹 ↾ 𝐷):𝐷–1-1-onto→𝐷)) |
14 | 3, 8, 13 | 3bitrd 307 | 1 ⊢ (𝐹 ∈ 𝑉 → ((𝐹 ↾ 𝐷) ∈ 𝑆 ↔ (𝐹 ↾ 𝐷):𝐷–1-1-onto→𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 {crab 3144 Vcvv 3496 ∖ cdif 3935 {csn 4569 ↾ cres 5559 –1-1-onto→wf1o 6356 ‘cfv 6357 Basecbs 16485 SymGrpcsymg 18497 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-tset 16586 df-efmnd 18036 df-symg 18498 |
This theorem is referenced by: symgfixelsi 18565 |
Copyright terms: Public domain | W3C validator |