symgfix2 - Metamath Proof Explorer

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Theorem symgfix2 19371

Description: If a permutation does not move a certain element of a set to a second element, there is a third element which is moved to the second element. (Contributed by AV, 2-Jan-2019.)

**Hypothesis**
Ref	Expression
symgfix2.p	⊢ 𝑃 = (Base‘(SymGrp‘𝑁))

**Assertion**
Ref	Expression
symgfix2	⊢ (𝐿 ∈ 𝑁 → (𝑄 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿}) → ∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄‘𝑘) = 𝐿))

Distinct variable groups: 𝑘,𝑁 𝑄,𝑘 𝑘,𝐾,𝑞 𝑘,𝐿,𝑞 𝑃,𝑞 𝑄,𝑞 Allowed substitution hints: 𝑃(𝑘) 𝑁(𝑞)

Proof of Theorem symgfix2 Dummy variable 𝑙 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldif 3957	. . 3 ⊢ (𝑄 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿}) ↔ (𝑄 ∈ 𝑃 ∧ ¬ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿}))
2		ianor 980	. . . . 5 ⊢ (¬ (𝑄 ∈ 𝑃 ∧ (𝑄‘𝐾) = 𝐿) ↔ (¬ 𝑄 ∈ 𝑃 ∨ ¬ (𝑄‘𝐾) = 𝐿))
3		fveq1 6896	. . . . . . 7 ⊢ (𝑞 = 𝑄 → (𝑞‘𝐾) = (𝑄‘𝐾))
4	3	eqeq1d 2730	. . . . . 6 ⊢ (𝑞 = 𝑄 → ((𝑞‘𝐾) = 𝐿 ↔ (𝑄‘𝐾) = 𝐿))
5	4	elrab 3682	. . . . 5 ⊢ (𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿} ↔ (𝑄 ∈ 𝑃 ∧ (𝑄‘𝐾) = 𝐿))
6	2, 5	xchnxbir 333	. . . 4 ⊢ (¬ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿} ↔ (¬ 𝑄 ∈ 𝑃 ∨ ¬ (𝑄‘𝐾) = 𝐿))
7	6	anbi2i 622	. . 3 ⊢ ((𝑄 ∈ 𝑃 ∧ ¬ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿}) ↔ (𝑄 ∈ 𝑃 ∧ (¬ 𝑄 ∈ 𝑃 ∨ ¬ (𝑄‘𝐾) = 𝐿)))
8	1, 7	bitri 275	. 2 ⊢ (𝑄 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿}) ↔ (𝑄 ∈ 𝑃 ∧ (¬ 𝑄 ∈ 𝑃 ∨ ¬ (𝑄‘𝐾) = 𝐿)))
9		pm2.21 123	. . . . 5 ⊢ (¬ 𝑄 ∈ 𝑃 → (𝑄 ∈ 𝑃 → (𝐿 ∈ 𝑁 → ∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄‘𝑘) = 𝐿)))
10		symgfix2.p	. . . . . . 7 ⊢ 𝑃 = (Base‘(SymGrp‘𝑁))
11	10	symgmov2 19342	. . . . . 6 ⊢ (𝑄 ∈ 𝑃 → ∀𝑙 ∈ 𝑁 ∃𝑘 ∈ 𝑁 (𝑄‘𝑘) = 𝑙)
12		eqeq2 2740	. . . . . . . . . . 11 ⊢ (𝑙 = 𝐿 → ((𝑄‘𝑘) = 𝑙 ↔ (𝑄‘𝑘) = 𝐿))
13	12	rexbidv 3175	. . . . . . . . . 10 ⊢ (𝑙 = 𝐿 → (∃𝑘 ∈ 𝑁 (𝑄‘𝑘) = 𝑙 ↔ ∃𝑘 ∈ 𝑁 (𝑄‘𝑘) = 𝐿))
14	13	rspcva 3607	. . . . . . . . 9 ⊢ ((𝐿 ∈ 𝑁 ∧ ∀𝑙 ∈ 𝑁 ∃𝑘 ∈ 𝑁 (𝑄‘𝑘) = 𝑙) → ∃𝑘 ∈ 𝑁 (𝑄‘𝑘) = 𝐿)
15		eqeq2 2740	. . . . . . . . . . . . . . . 16 ⊢ (𝐿 = (𝑄‘𝑘) → ((𝑄‘𝐾) = 𝐿 ↔ (𝑄‘𝐾) = (𝑄‘𝑘)))
16	15	eqcoms 2736	. . . . . . . . . . . . . . 15 ⊢ ((𝑄‘𝑘) = 𝐿 → ((𝑄‘𝐾) = 𝐿 ↔ (𝑄‘𝐾) = (𝑄‘𝑘)))
17	16	notbid 318	. . . . . . . . . . . . . 14 ⊢ ((𝑄‘𝑘) = 𝐿 → (¬ (𝑄‘𝐾) = 𝐿 ↔ ¬ (𝑄‘𝐾) = (𝑄‘𝑘)))
18		fveq2 6897	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 = 𝑘 → (𝑄‘𝐾) = (𝑄‘𝑘))
19	18	eqcoms 2736	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝐾 → (𝑄‘𝐾) = (𝑄‘𝑘))
20	19	necon3bi 2964	. . . . . . . . . . . . . 14 ⊢ (¬ (𝑄‘𝐾) = (𝑄‘𝑘) → 𝑘 ≠ 𝐾)
21	17, 20	biimtrdi 252	. . . . . . . . . . . . 13 ⊢ ((𝑄‘𝑘) = 𝐿 → (¬ (𝑄‘𝐾) = 𝐿 → 𝑘 ≠ 𝐾))
22	21	com12 32	. . . . . . . . . . . 12 ⊢ (¬ (𝑄‘𝐾) = 𝐿 → ((𝑄‘𝑘) = 𝐿 → 𝑘 ≠ 𝐾))
23	22	pm4.71rd 562	. . . . . . . . . . 11 ⊢ (¬ (𝑄‘𝐾) = 𝐿 → ((𝑄‘𝑘) = 𝐿 ↔ (𝑘 ≠ 𝐾 ∧ (𝑄‘𝑘) = 𝐿)))
24	23	rexbidv 3175	. . . . . . . . . 10 ⊢ (¬ (𝑄‘𝐾) = 𝐿 → (∃𝑘 ∈ 𝑁 (𝑄‘𝑘) = 𝐿 ↔ ∃𝑘 ∈ 𝑁 (𝑘 ≠ 𝐾 ∧ (𝑄‘𝑘) = 𝐿)))
25		rexdifsn 4798	. . . . . . . . . 10 ⊢ (∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄‘𝑘) = 𝐿 ↔ ∃𝑘 ∈ 𝑁 (𝑘 ≠ 𝐾 ∧ (𝑄‘𝑘) = 𝐿))
26	24, 25	bitr4di 289	. . . . . . . . 9 ⊢ (¬ (𝑄‘𝐾) = 𝐿 → (∃𝑘 ∈ 𝑁 (𝑄‘𝑘) = 𝐿 ↔ ∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄‘𝑘) = 𝐿))
27	14, 26	syl5ibcom 244	. . . . . . . 8 ⊢ ((𝐿 ∈ 𝑁 ∧ ∀𝑙 ∈ 𝑁 ∃𝑘 ∈ 𝑁 (𝑄‘𝑘) = 𝑙) → (¬ (𝑄‘𝐾) = 𝐿 → ∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄‘𝑘) = 𝐿))
28	27	ex 412	. . . . . . 7 ⊢ (𝐿 ∈ 𝑁 → (∀𝑙 ∈ 𝑁 ∃𝑘 ∈ 𝑁 (𝑄‘𝑘) = 𝑙 → (¬ (𝑄‘𝐾) = 𝐿 → ∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄‘𝑘) = 𝐿)))
29	28	com13 88	. . . . . 6 ⊢ (¬ (𝑄‘𝐾) = 𝐿 → (∀𝑙 ∈ 𝑁 ∃𝑘 ∈ 𝑁 (𝑄‘𝑘) = 𝑙 → (𝐿 ∈ 𝑁 → ∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄‘𝑘) = 𝐿)))
30	11, 29	syl5 34	. . . . 5 ⊢ (¬ (𝑄‘𝐾) = 𝐿 → (𝑄 ∈ 𝑃 → (𝐿 ∈ 𝑁 → ∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄‘𝑘) = 𝐿)))
31	9, 30	jaoi 856	. . . 4 ⊢ ((¬ 𝑄 ∈ 𝑃 ∨ ¬ (𝑄‘𝐾) = 𝐿) → (𝑄 ∈ 𝑃 → (𝐿 ∈ 𝑁 → ∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄‘𝑘) = 𝐿)))
32	31	com13 88	. . 3 ⊢ (𝐿 ∈ 𝑁 → (𝑄 ∈ 𝑃 → ((¬ 𝑄 ∈ 𝑃 ∨ ¬ (𝑄‘𝐾) = 𝐿) → ∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄‘𝑘) = 𝐿)))
33	32	impd 410	. 2 ⊢ (𝐿 ∈ 𝑁 → ((𝑄 ∈ 𝑃 ∧ (¬ 𝑄 ∈ 𝑃 ∨ ¬ (𝑄‘𝐾) = 𝐿)) → ∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄‘𝑘) = 𝐿))
34	8, 33	biimtrid 241	1 ⊢ (𝐿 ∈ 𝑁 → (𝑄 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿}) → ∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄‘𝑘) = 𝐿))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 846 = wceq 1534 ∈ wcel 2099 ≠ wne 2937 ∀wral 3058 ∃wrex 3067 {crab 3429 ∖ cdif 3944 {csn 4629 ‘cfv 6548 Basecbs 17180 SymGrpcsymg 19321

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 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-uz 12854 df-fz 13518 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-tset 17252 df-efmnd 18821 df-symg 19322

This theorem is referenced by: (None)

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