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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 3492 | . . 3 ⊢ (𝐹 ∈ 𝐵 → 𝐹 ∈ V) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐹 ∈ 𝐵 → 𝐹 ∈ V)) |
3 | f1of 6833 | . . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → 𝐹:𝐴⟶𝐴) | |
4 | fex 7230 | . . . 4 ⊢ ((𝐹:𝐴⟶𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V) | |
5 | 4 | expcom 413 | . . 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 19282 | . . 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 379 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐹 ∈ 𝐵 ↔ 𝐹:𝐴–1-1-onto→𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1540 ∈ wcel 2105 Vcvv 3473 ⟶wf 6539 –1-1-onto→wf1o 6542 ‘cfv 6543 Basecbs 17149 SymGrpcsymg 19276 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 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 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-er 8707 df-map 8826 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12478 df-z 12564 df-uz 12828 df-fz 13490 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-tset 17221 df-efmnd 18787 df-symg 19277 |
This theorem is referenced by: idresperm 19295 symgpssefmnd 19305 symggrp 19310 galactghm 19314 lactghmga 19315 symgextsymg 19334 mdetunilem7 22341 mdetunilem8 22342 symgtgp 23831 cycpmconjv 32572 cycpmrn 32573 cycpmconjs 32586 cyc3conja 32587 |
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