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Mirrors > Home > MPE Home > Th. List > symgbasfi | Structured version Visualization version GIF version |
Description: The symmetric group on a finite index set is finite. (Contributed by SO, 9-Jul-2018.) |
Ref | Expression |
---|---|
symgbas.1 | ⊢ 𝐺 = (SymGrp‘𝐴) |
symgbas.2 | ⊢ 𝐵 = (Base‘𝐺) |
Ref | Expression |
---|---|
symgbasfi | ⊢ (𝐴 ∈ Fin → 𝐵 ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapfi 8145 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ∈ Fin) → (𝐴 ↑𝑚 𝐴) ∈ Fin) | |
2 | 1 | anidms 675 | . 2 ⊢ (𝐴 ∈ Fin → (𝐴 ↑𝑚 𝐴) ∈ Fin) |
3 | symgbas.1 | . . . . 5 ⊢ 𝐺 = (SymGrp‘𝐴) | |
4 | symgbas.2 | . . . . 5 ⊢ 𝐵 = (Base‘𝐺) | |
5 | 3, 4 | symgbas 17623 | . . . 4 ⊢ 𝐵 = {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐴} |
6 | f1of 6050 | . . . . 5 ⊢ (𝑓:𝐴–1-1-onto→𝐴 → 𝑓:𝐴⟶𝐴) | |
7 | 6 | ss2abi 3637 | . . . 4 ⊢ {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐴} ⊆ {𝑓 ∣ 𝑓:𝐴⟶𝐴} |
8 | 5, 7 | eqsstri 3598 | . . 3 ⊢ 𝐵 ⊆ {𝑓 ∣ 𝑓:𝐴⟶𝐴} |
9 | mapvalg 7754 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ∈ Fin) → (𝐴 ↑𝑚 𝐴) = {𝑓 ∣ 𝑓:𝐴⟶𝐴}) | |
10 | 9 | anidms 675 | . . 3 ⊢ (𝐴 ∈ Fin → (𝐴 ↑𝑚 𝐴) = {𝑓 ∣ 𝑓:𝐴⟶𝐴}) |
11 | 8, 10 | syl5sseqr 3617 | . 2 ⊢ (𝐴 ∈ Fin → 𝐵 ⊆ (𝐴 ↑𝑚 𝐴)) |
12 | ssfi 8065 | . 2 ⊢ (((𝐴 ↑𝑚 𝐴) ∈ Fin ∧ 𝐵 ⊆ (𝐴 ↑𝑚 𝐴)) → 𝐵 ∈ Fin) | |
13 | 2, 11, 12 | syl2anc 691 | 1 ⊢ (𝐴 ∈ Fin → 𝐵 ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 {cab 2596 ⊆ wss 3540 ⟶wf 5800 –1-1-onto→wf1o 5803 ‘cfv 5804 (class class class)co 6549 ↑𝑚 cmap 7744 Fincfn 7841 Basecbs 15695 SymGrpcsymg 17620 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-plusg 15781 df-tset 15787 df-symg 17621 |
This theorem is referenced by: mdetleib2 20213 mdetf 20220 mdetrlin 20227 mdetrsca 20228 mdetralt 20233 m2detleib 20256 smadiadetlem3 20293 smadiadet 20295 mdetpmtr1 29217 |
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