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Mirrors > Home > MPE Home > Th. List > elsymgbas | Structured version Visualization version GIF version |
Description: Two ways of saying a function is a 1-1-onto mapping of A to itself. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by Mario Carneiro, 28-Jan-2015.) |
Ref | Expression |
---|---|
symgbas.1 | ⊢ 𝐺 = (SymGrp‘𝐴) |
symgbas.2 | ⊢ 𝐵 = (Base‘𝐺) |
Ref | Expression |
---|---|
elsymgbas | ⊢ (𝐴 ∈ 𝑉 → (𝐹 ∈ 𝐵 ↔ 𝐹:𝐴–1-1-onto→𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3493 | . . 3 ⊢ (𝐹 ∈ 𝐵 → 𝐹 ∈ V) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐹 ∈ 𝐵 → 𝐹 ∈ V)) |
3 | f1of 6831 | . . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → 𝐹:𝐴⟶𝐴) | |
4 | fex 7225 | . . . 4 ⊢ ((𝐹:𝐴⟶𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V) | |
5 | 4 | expcom 415 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐹:𝐴⟶𝐴 → 𝐹 ∈ V)) |
6 | 3, 5 | syl5 34 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐹:𝐴–1-1-onto→𝐴 → 𝐹 ∈ V)) |
7 | symgbas.1 | . . . 4 ⊢ 𝐺 = (SymGrp‘𝐴) | |
8 | symgbas.2 | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
9 | 7, 8 | elsymgbas2 19235 | . . 3 ⊢ (𝐹 ∈ V → (𝐹 ∈ 𝐵 ↔ 𝐹:𝐴–1-1-onto→𝐴)) |
10 | 9 | a1i 11 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐹 ∈ V → (𝐹 ∈ 𝐵 ↔ 𝐹:𝐴–1-1-onto→𝐴))) |
11 | 2, 6, 10 | pm5.21ndd 381 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐹 ∈ 𝐵 ↔ 𝐹:𝐴–1-1-onto→𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∈ wcel 2107 Vcvv 3475 ⟶wf 6537 –1-1-onto→wf1o 6540 ‘cfv 6541 Basecbs 17141 SymGrpcsymg 19229 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-1st 7972 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-uz 12820 df-fz 13482 df-struct 17077 df-sets 17094 df-slot 17112 df-ndx 17124 df-base 17142 df-ress 17171 df-plusg 17207 df-tset 17213 df-efmnd 18747 df-symg 19230 |
This theorem is referenced by: idresperm 19248 symgpssefmnd 19258 symggrp 19263 galactghm 19267 lactghmga 19268 symgextsymg 19287 mdetunilem7 22112 mdetunilem8 22113 symgtgp 23602 cycpmconjv 32289 cycpmrn 32290 cycpmconjs 32303 cyc3conja 32304 |
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