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| Mirrors > Home > MPE Home > Th. List > symg1bas | Structured version Visualization version GIF version | ||
| Description: The symmetric group on a singleton is the symmetric group S1 consisting of the identity only. (Contributed by AV, 9-Dec-2018.) |
| Ref | Expression |
|---|---|
| symg1bas.1 | ⊢ 𝐺 = (SymGrp‘𝐴) |
| symg1bas.2 | ⊢ 𝐵 = (Base‘𝐺) |
| symg1bas.0 | ⊢ 𝐴 = {𝐼} |
| Ref | Expression |
|---|---|
| symg1bas | ⊢ (𝐼 ∈ 𝑉 → 𝐵 = {{〈𝐼, 𝐼〉}}) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | symg1bas.1 | . . 3 ⊢ 𝐺 = (SymGrp‘𝐴) | |
| 2 | symg1bas.2 | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
| 3 | 1, 2 | symgbas 19389 | . 2 ⊢ 𝐵 = {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐴} |
| 4 | symg1bas.0 | . . . . . 6 ⊢ 𝐴 = {𝐼} | |
| 5 | eqidd 2738 | . . . . . . 7 ⊢ (𝐴 = {𝐼} → 𝑝 = 𝑝) | |
| 6 | id 22 | . . . . . . 7 ⊢ (𝐴 = {𝐼} → 𝐴 = {𝐼}) | |
| 7 | 5, 6, 6 | f1oeq123d 6842 | . . . . . 6 ⊢ (𝐴 = {𝐼} → (𝑝:𝐴–1-1-onto→𝐴 ↔ 𝑝:{𝐼}–1-1-onto→{𝐼})) |
| 8 | 4, 7 | ax-mp 5 | . . . . 5 ⊢ (𝑝:𝐴–1-1-onto→𝐴 ↔ 𝑝:{𝐼}–1-1-onto→{𝐼}) |
| 9 | f1of 6848 | . . . . . . 7 ⊢ (𝑝:{𝐼}–1-1-onto→{𝐼} → 𝑝:{𝐼}⟶{𝐼}) | |
| 10 | fsng 7157 | . . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐼 ∈ 𝑉) → (𝑝:{𝐼}⟶{𝐼} ↔ 𝑝 = {〈𝐼, 𝐼〉})) | |
| 11 | 10 | anidms 566 | . . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → (𝑝:{𝐼}⟶{𝐼} ↔ 𝑝 = {〈𝐼, 𝐼〉})) |
| 12 | 9, 11 | imbitrid 244 | . . . . . 6 ⊢ (𝐼 ∈ 𝑉 → (𝑝:{𝐼}–1-1-onto→{𝐼} → 𝑝 = {〈𝐼, 𝐼〉})) |
| 13 | f1osng 6889 | . . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐼 ∈ 𝑉) → {〈𝐼, 𝐼〉}:{𝐼}–1-1-onto→{𝐼}) | |
| 14 | 13 | anidms 566 | . . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → {〈𝐼, 𝐼〉}:{𝐼}–1-1-onto→{𝐼}) |
| 15 | f1oeq1 6836 | . . . . . . 7 ⊢ (𝑝 = {〈𝐼, 𝐼〉} → (𝑝:{𝐼}–1-1-onto→{𝐼} ↔ {〈𝐼, 𝐼〉}:{𝐼}–1-1-onto→{𝐼})) | |
| 16 | 14, 15 | syl5ibrcom 247 | . . . . . 6 ⊢ (𝐼 ∈ 𝑉 → (𝑝 = {〈𝐼, 𝐼〉} → 𝑝:{𝐼}–1-1-onto→{𝐼})) |
| 17 | 12, 16 | impbid 212 | . . . . 5 ⊢ (𝐼 ∈ 𝑉 → (𝑝:{𝐼}–1-1-onto→{𝐼} ↔ 𝑝 = {〈𝐼, 𝐼〉})) |
| 18 | 8, 17 | bitrid 283 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → (𝑝:𝐴–1-1-onto→𝐴 ↔ 𝑝 = {〈𝐼, 𝐼〉})) |
| 19 | vex 3484 | . . . . 5 ⊢ 𝑝 ∈ V | |
| 20 | f1oeq1 6836 | . . . . 5 ⊢ (𝑓 = 𝑝 → (𝑓:𝐴–1-1-onto→𝐴 ↔ 𝑝:𝐴–1-1-onto→𝐴)) | |
| 21 | 19, 20 | elab 3679 | . . . 4 ⊢ (𝑝 ∈ {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐴} ↔ 𝑝:𝐴–1-1-onto→𝐴) |
| 22 | velsn 4642 | . . . 4 ⊢ (𝑝 ∈ {{〈𝐼, 𝐼〉}} ↔ 𝑝 = {〈𝐼, 𝐼〉}) | |
| 23 | 18, 21, 22 | 3bitr4g 314 | . . 3 ⊢ (𝐼 ∈ 𝑉 → (𝑝 ∈ {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐴} ↔ 𝑝 ∈ {{〈𝐼, 𝐼〉}})) |
| 24 | 23 | eqrdv 2735 | . 2 ⊢ (𝐼 ∈ 𝑉 → {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐴} = {{〈𝐼, 𝐼〉}}) |
| 25 | 3, 24 | eqtrid 2789 | 1 ⊢ (𝐼 ∈ 𝑉 → 𝐵 = {{〈𝐼, 𝐼〉}}) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 {cab 2714 {csn 4626 〈cop 4632 ⟶wf 6557 –1-1-onto→wf1o 6560 ‘cfv 6561 Basecbs 17247 SymGrpcsymg 19386 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-tset 17316 df-efmnd 18882 df-symg 19387 |
| This theorem is referenced by: symg2bas 19410 snsymgefmndeq 19412 psgnsn 19538 m1detdiag 22603 |
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