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Mirrors > Home > MPE Home > Th. List > symgbas | Structured version Visualization version GIF version |
Description: The base set of the symmetric group. (Contributed by Mario Carneiro, 12-Jan-2015.) (Proof shortened by AV, 29-Mar-2024.) |
Ref | Expression |
---|---|
symgbas.1 | ⊢ 𝐺 = (SymGrp‘𝐴) |
symgbas.2 | ⊢ 𝐵 = (Base‘𝐺) |
Ref | Expression |
---|---|
symgbas | ⊢ 𝐵 = {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | symgbas.1 | . . . . 5 ⊢ 𝐺 = (SymGrp‘𝐴) | |
2 | eqid 2726 | . . . . 5 ⊢ {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴} = {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴} | |
3 | 1, 2 | symgval 19368 | . . . 4 ⊢ 𝐺 = ((EndoFMnd‘𝐴) ↾s {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴}) |
4 | 3 | eqcomi 2735 | . . 3 ⊢ ((EndoFMnd‘𝐴) ↾s {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴}) = 𝐺 |
5 | 4 | fveq2i 6906 | . 2 ⊢ (Base‘((EndoFMnd‘𝐴) ↾s {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴})) = (Base‘𝐺) |
6 | f1of 6845 | . . . . 5 ⊢ (𝑥:𝐴–1-1-onto→𝐴 → 𝑥:𝐴⟶𝐴) | |
7 | 6 | ss2abi 4062 | . . . 4 ⊢ {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴} ⊆ {𝑥 ∣ 𝑥:𝐴⟶𝐴} |
8 | eqid 2726 | . . . . 5 ⊢ (EndoFMnd‘𝐴) = (EndoFMnd‘𝐴) | |
9 | eqid 2726 | . . . . 5 ⊢ (Base‘(EndoFMnd‘𝐴)) = (Base‘(EndoFMnd‘𝐴)) | |
10 | 8, 9 | efmndbasabf 18864 | . . . 4 ⊢ (Base‘(EndoFMnd‘𝐴)) = {𝑥 ∣ 𝑥:𝐴⟶𝐴} |
11 | 7, 10 | sseqtrri 4017 | . . 3 ⊢ {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴} ⊆ (Base‘(EndoFMnd‘𝐴)) |
12 | eqid 2726 | . . . 4 ⊢ ((EndoFMnd‘𝐴) ↾s {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴}) = ((EndoFMnd‘𝐴) ↾s {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴}) | |
13 | 12, 9 | ressbas2 17253 | . . 3 ⊢ ({𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴} ⊆ (Base‘(EndoFMnd‘𝐴)) → {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴} = (Base‘((EndoFMnd‘𝐴) ↾s {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴}))) |
14 | 11, 13 | ax-mp 5 | . 2 ⊢ {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴} = (Base‘((EndoFMnd‘𝐴) ↾s {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴})) |
15 | symgbas.2 | . 2 ⊢ 𝐵 = (Base‘𝐺) | |
16 | 5, 14, 15 | 3eqtr4ri 2765 | 1 ⊢ 𝐵 = {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 {cab 2703 ⊆ wss 3947 ⟶wf 6552 –1-1-onto→wf1o 6555 ‘cfv 6556 (class class class)co 7426 Basecbs 17215 ↾s cress 17244 EndoFMndcefmnd 18860 SymGrpcsymg 19366 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5306 ax-nul 5313 ax-pow 5371 ax-pr 5435 ax-un 7748 ax-cnex 11216 ax-resscn 11217 ax-1cn 11218 ax-icn 11219 ax-addcl 11220 ax-addrcl 11221 ax-mulcl 11222 ax-mulrcl 11223 ax-mulcom 11224 ax-addass 11225 ax-mulass 11226 ax-distr 11227 ax-i2m1 11228 ax-1ne0 11229 ax-1rid 11230 ax-rnegex 11231 ax-rrecex 11232 ax-cnre 11233 ax-pre-lttri 11234 ax-pre-lttrn 11235 ax-pre-ltadd 11236 ax-pre-mulgt0 11237 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-uni 4916 df-iun 5005 df-br 5156 df-opab 5218 df-mpt 5239 df-tr 5273 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5639 df-we 5641 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6314 df-ord 6381 df-on 6382 df-lim 6383 df-suc 6384 df-iota 6508 df-fun 6558 df-fn 6559 df-f 6560 df-f1 6561 df-fo 6562 df-f1o 6563 df-fv 6564 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8005 df-2nd 8006 df-frecs 8298 df-wrecs 8329 df-recs 8403 df-rdg 8442 df-1o 8498 df-er 8736 df-map 8859 df-en 8977 df-dom 8978 df-sdom 8979 df-fin 8980 df-pnf 11302 df-mnf 11303 df-xr 11304 df-ltxr 11305 df-le 11306 df-sub 11498 df-neg 11499 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 12613 df-uz 12877 df-fz 13541 df-struct 17151 df-sets 17168 df-slot 17186 df-ndx 17198 df-base 17216 df-ress 17245 df-plusg 17281 df-tset 17287 df-efmnd 18861 df-symg 19367 |
This theorem is referenced by: symgbasexOLD 19371 elsymgbas2 19372 symghash 19377 symgbasfi 19378 symgressbas 19381 symgbas0 19388 symg1bas 19390 symgvalstruct 19396 symgvalstructOLD 19397 symgsubmefmnd 19398 symgtset 19399 |
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