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Theorem symgextf1 18191
Description: The extension of a permutation, fixing the additional element, is a 1-1 function. (Contributed by AV, 6-Jan-2019.)
Hypotheses
Ref Expression
symgext.s 𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))
symgext.e 𝐸 = (𝑥𝑁 ↦ if(𝑥 = 𝐾, 𝐾, (𝑍𝑥)))
Assertion
Ref Expression
symgextf1 ((𝐾𝑁𝑍𝑆) → 𝐸:𝑁1-1𝑁)
Distinct variable groups:   𝑥,𝐾   𝑥,𝑁   𝑥,𝑆   𝑥,𝑍
Allowed substitution hint:   𝐸(𝑥)

Proof of Theorem symgextf1
Dummy variables 𝑦 𝑧 𝑖 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 symgext.s . . 3 𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))
2 symgext.e . . 3 𝐸 = (𝑥𝑁 ↦ if(𝑥 = 𝐾, 𝐾, (𝑍𝑥)))
31, 2symgextf 18187 . 2 ((𝐾𝑁𝑍𝑆) → 𝐸:𝑁𝑁)
4 difsnid 4559 . . . . . . . 8 (𝐾𝑁 → ((𝑁 ∖ {𝐾}) ∪ {𝐾}) = 𝑁)
54eqcomd 2831 . . . . . . 7 (𝐾𝑁𝑁 = ((𝑁 ∖ {𝐾}) ∪ {𝐾}))
65eleq2d 2892 . . . . . 6 (𝐾𝑁 → (𝑦𝑁𝑦 ∈ ((𝑁 ∖ {𝐾}) ∪ {𝐾})))
75eleq2d 2892 . . . . . 6 (𝐾𝑁 → (𝑧𝑁𝑧 ∈ ((𝑁 ∖ {𝐾}) ∪ {𝐾})))
86, 7anbi12d 626 . . . . 5 (𝐾𝑁 → ((𝑦𝑁𝑧𝑁) ↔ (𝑦 ∈ ((𝑁 ∖ {𝐾}) ∪ {𝐾}) ∧ 𝑧 ∈ ((𝑁 ∖ {𝐾}) ∪ {𝐾}))))
98adantr 474 . . . 4 ((𝐾𝑁𝑍𝑆) → ((𝑦𝑁𝑧𝑁) ↔ (𝑦 ∈ ((𝑁 ∖ {𝐾}) ∪ {𝐾}) ∧ 𝑧 ∈ ((𝑁 ∖ {𝐾}) ∪ {𝐾}))))
10 elun 3980 . . . . . 6 (𝑦 ∈ ((𝑁 ∖ {𝐾}) ∪ {𝐾}) ↔ (𝑦 ∈ (𝑁 ∖ {𝐾}) ∨ 𝑦 ∈ {𝐾}))
11 elun 3980 . . . . . 6 (𝑧 ∈ ((𝑁 ∖ {𝐾}) ∪ {𝐾}) ↔ (𝑧 ∈ (𝑁 ∖ {𝐾}) ∨ 𝑧 ∈ {𝐾}))
121, 2symgextfv 18188 . . . . . . . . . . . . 13 ((𝐾𝑁𝑍𝑆) → (𝑦 ∈ (𝑁 ∖ {𝐾}) → (𝐸𝑦) = (𝑍𝑦)))
1312com12 32 . . . . . . . . . . . 12 (𝑦 ∈ (𝑁 ∖ {𝐾}) → ((𝐾𝑁𝑍𝑆) → (𝐸𝑦) = (𝑍𝑦)))
1413adantr 474 . . . . . . . . . . 11 ((𝑦 ∈ (𝑁 ∖ {𝐾}) ∧ 𝑧 ∈ (𝑁 ∖ {𝐾})) → ((𝐾𝑁𝑍𝑆) → (𝐸𝑦) = (𝑍𝑦)))
1514imp 397 . . . . . . . . . 10 (((𝑦 ∈ (𝑁 ∖ {𝐾}) ∧ 𝑧 ∈ (𝑁 ∖ {𝐾})) ∧ (𝐾𝑁𝑍𝑆)) → (𝐸𝑦) = (𝑍𝑦))
161, 2symgextfv 18188 . . . . . . . . . . . . 13 ((𝐾𝑁𝑍𝑆) → (𝑧 ∈ (𝑁 ∖ {𝐾}) → (𝐸𝑧) = (𝑍𝑧)))
1716com12 32 . . . . . . . . . . . 12 (𝑧 ∈ (𝑁 ∖ {𝐾}) → ((𝐾𝑁𝑍𝑆) → (𝐸𝑧) = (𝑍𝑧)))
1817adantl 475 . . . . . . . . . . 11 ((𝑦 ∈ (𝑁 ∖ {𝐾}) ∧ 𝑧 ∈ (𝑁 ∖ {𝐾})) → ((𝐾𝑁𝑍𝑆) → (𝐸𝑧) = (𝑍𝑧)))
1918imp 397 . . . . . . . . . 10 (((𝑦 ∈ (𝑁 ∖ {𝐾}) ∧ 𝑧 ∈ (𝑁 ∖ {𝐾})) ∧ (𝐾𝑁𝑍𝑆)) → (𝐸𝑧) = (𝑍𝑧))
2015, 19eqeq12d 2840 . . . . . . . . 9 (((𝑦 ∈ (𝑁 ∖ {𝐾}) ∧ 𝑧 ∈ (𝑁 ∖ {𝐾})) ∧ (𝐾𝑁𝑍𝑆)) → ((𝐸𝑦) = (𝐸𝑧) ↔ (𝑍𝑦) = (𝑍𝑧)))
21 eqid 2825 . . . . . . . . . . . . 13 (SymGrp‘(𝑁 ∖ {𝐾})) = (SymGrp‘(𝑁 ∖ {𝐾}))
2221, 1symgbasf1o 18153 . . . . . . . . . . . 12 (𝑍𝑆𝑍:(𝑁 ∖ {𝐾})–1-1-onto→(𝑁 ∖ {𝐾}))
23 f1of1 6377 . . . . . . . . . . . 12 (𝑍:(𝑁 ∖ {𝐾})–1-1-onto→(𝑁 ∖ {𝐾}) → 𝑍:(𝑁 ∖ {𝐾})–1-1→(𝑁 ∖ {𝐾}))
24 dff13 6767 . . . . . . . . . . . . 13 (𝑍:(𝑁 ∖ {𝐾})–1-1→(𝑁 ∖ {𝐾}) ↔ (𝑍:(𝑁 ∖ {𝐾})⟶(𝑁 ∖ {𝐾}) ∧ ∀𝑖 ∈ (𝑁 ∖ {𝐾})∀𝑗 ∈ (𝑁 ∖ {𝐾})((𝑍𝑖) = (𝑍𝑗) → 𝑖 = 𝑗)))
25 fveqeq2 6442 . . . . . . . . . . . . . . . . . 18 (𝑖 = 𝑦 → ((𝑍𝑖) = (𝑍𝑗) ↔ (𝑍𝑦) = (𝑍𝑗)))
26 equequ1 2131 . . . . . . . . . . . . . . . . . 18 (𝑖 = 𝑦 → (𝑖 = 𝑗𝑦 = 𝑗))
2725, 26imbi12d 336 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝑦 → (((𝑍𝑖) = (𝑍𝑗) → 𝑖 = 𝑗) ↔ ((𝑍𝑦) = (𝑍𝑗) → 𝑦 = 𝑗)))
28 fveq2 6433 . . . . . . . . . . . . . . . . . . 19 (𝑗 = 𝑧 → (𝑍𝑗) = (𝑍𝑧))
2928eqeq2d 2835 . . . . . . . . . . . . . . . . . 18 (𝑗 = 𝑧 → ((𝑍𝑦) = (𝑍𝑗) ↔ (𝑍𝑦) = (𝑍𝑧)))
30 equequ2 2132 . . . . . . . . . . . . . . . . . 18 (𝑗 = 𝑧 → (𝑦 = 𝑗𝑦 = 𝑧))
3129, 30imbi12d 336 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑧 → (((𝑍𝑦) = (𝑍𝑗) → 𝑦 = 𝑗) ↔ ((𝑍𝑦) = (𝑍𝑧) → 𝑦 = 𝑧)))
3227, 31rspc2va 3540 . . . . . . . . . . . . . . . 16 (((𝑦 ∈ (𝑁 ∖ {𝐾}) ∧ 𝑧 ∈ (𝑁 ∖ {𝐾})) ∧ ∀𝑖 ∈ (𝑁 ∖ {𝐾})∀𝑗 ∈ (𝑁 ∖ {𝐾})((𝑍𝑖) = (𝑍𝑗) → 𝑖 = 𝑗)) → ((𝑍𝑦) = (𝑍𝑧) → 𝑦 = 𝑧))
3332expcom 404 . . . . . . . . . . . . . . 15 (∀𝑖 ∈ (𝑁 ∖ {𝐾})∀𝑗 ∈ (𝑁 ∖ {𝐾})((𝑍𝑖) = (𝑍𝑗) → 𝑖 = 𝑗) → ((𝑦 ∈ (𝑁 ∖ {𝐾}) ∧ 𝑧 ∈ (𝑁 ∖ {𝐾})) → ((𝑍𝑦) = (𝑍𝑧) → 𝑦 = 𝑧)))
3433a1d 25 . . . . . . . . . . . . . 14 (∀𝑖 ∈ (𝑁 ∖ {𝐾})∀𝑗 ∈ (𝑁 ∖ {𝐾})((𝑍𝑖) = (𝑍𝑗) → 𝑖 = 𝑗) → (𝐾𝑁 → ((𝑦 ∈ (𝑁 ∖ {𝐾}) ∧ 𝑧 ∈ (𝑁 ∖ {𝐾})) → ((𝑍𝑦) = (𝑍𝑧) → 𝑦 = 𝑧))))
3534adantl 475 . . . . . . . . . . . . 13 ((𝑍:(𝑁 ∖ {𝐾})⟶(𝑁 ∖ {𝐾}) ∧ ∀𝑖 ∈ (𝑁 ∖ {𝐾})∀𝑗 ∈ (𝑁 ∖ {𝐾})((𝑍𝑖) = (𝑍𝑗) → 𝑖 = 𝑗)) → (𝐾𝑁 → ((𝑦 ∈ (𝑁 ∖ {𝐾}) ∧ 𝑧 ∈ (𝑁 ∖ {𝐾})) → ((𝑍𝑦) = (𝑍𝑧) → 𝑦 = 𝑧))))
3624, 35sylbi 209 . . . . . . . . . . . 12 (𝑍:(𝑁 ∖ {𝐾})–1-1→(𝑁 ∖ {𝐾}) → (𝐾𝑁 → ((𝑦 ∈ (𝑁 ∖ {𝐾}) ∧ 𝑧 ∈ (𝑁 ∖ {𝐾})) → ((𝑍𝑦) = (𝑍𝑧) → 𝑦 = 𝑧))))
3722, 23, 363syl 18 . . . . . . . . . . 11 (𝑍𝑆 → (𝐾𝑁 → ((𝑦 ∈ (𝑁 ∖ {𝐾}) ∧ 𝑧 ∈ (𝑁 ∖ {𝐾})) → ((𝑍𝑦) = (𝑍𝑧) → 𝑦 = 𝑧))))
3837impcom 398 . . . . . . . . . 10 ((𝐾𝑁𝑍𝑆) → ((𝑦 ∈ (𝑁 ∖ {𝐾}) ∧ 𝑧 ∈ (𝑁 ∖ {𝐾})) → ((𝑍𝑦) = (𝑍𝑧) → 𝑦 = 𝑧)))
3938impcom 398 . . . . . . . . 9 (((𝑦 ∈ (𝑁 ∖ {𝐾}) ∧ 𝑧 ∈ (𝑁 ∖ {𝐾})) ∧ (𝐾𝑁𝑍𝑆)) → ((𝑍𝑦) = (𝑍𝑧) → 𝑦 = 𝑧))
4020, 39sylbid 232 . . . . . . . 8 (((𝑦 ∈ (𝑁 ∖ {𝐾}) ∧ 𝑧 ∈ (𝑁 ∖ {𝐾})) ∧ (𝐾𝑁𝑍𝑆)) → ((𝐸𝑦) = (𝐸𝑧) → 𝑦 = 𝑧))
4140ex 403 . . . . . . 7 ((𝑦 ∈ (𝑁 ∖ {𝐾}) ∧ 𝑧 ∈ (𝑁 ∖ {𝐾})) → ((𝐾𝑁𝑍𝑆) → ((𝐸𝑦) = (𝐸𝑧) → 𝑦 = 𝑧)))
421, 2symgextf1lem 18190 . . . . . . . . 9 ((𝐾𝑁𝑍𝑆) → ((𝑧 ∈ (𝑁 ∖ {𝐾}) ∧ 𝑦 ∈ {𝐾}) → (𝐸𝑧) ≠ (𝐸𝑦)))
43 eqneqall 3010 . . . . . . . . . . 11 ((𝐸𝑧) = (𝐸𝑦) → ((𝐸𝑧) ≠ (𝐸𝑦) → 𝑦 = 𝑧))
4443eqcoms 2833 . . . . . . . . . 10 ((𝐸𝑦) = (𝐸𝑧) → ((𝐸𝑧) ≠ (𝐸𝑦) → 𝑦 = 𝑧))
4544com12 32 . . . . . . . . 9 ((𝐸𝑧) ≠ (𝐸𝑦) → ((𝐸𝑦) = (𝐸𝑧) → 𝑦 = 𝑧))
4642, 45syl6com 37 . . . . . . . 8 ((𝑧 ∈ (𝑁 ∖ {𝐾}) ∧ 𝑦 ∈ {𝐾}) → ((𝐾𝑁𝑍𝑆) → ((𝐸𝑦) = (𝐸𝑧) → 𝑦 = 𝑧)))
4746ancoms 452 . . . . . . 7 ((𝑦 ∈ {𝐾} ∧ 𝑧 ∈ (𝑁 ∖ {𝐾})) → ((𝐾𝑁𝑍𝑆) → ((𝐸𝑦) = (𝐸𝑧) → 𝑦 = 𝑧)))
481, 2symgextf1lem 18190 . . . . . . . 8 ((𝐾𝑁𝑍𝑆) → ((𝑦 ∈ (𝑁 ∖ {𝐾}) ∧ 𝑧 ∈ {𝐾}) → (𝐸𝑦) ≠ (𝐸𝑧)))
49 eqneqall 3010 . . . . . . . . 9 ((𝐸𝑦) = (𝐸𝑧) → ((𝐸𝑦) ≠ (𝐸𝑧) → 𝑦 = 𝑧))
5049com12 32 . . . . . . . 8 ((𝐸𝑦) ≠ (𝐸𝑧) → ((𝐸𝑦) = (𝐸𝑧) → 𝑦 = 𝑧))
5148, 50syl6com 37 . . . . . . 7 ((𝑦 ∈ (𝑁 ∖ {𝐾}) ∧ 𝑧 ∈ {𝐾}) → ((𝐾𝑁𝑍𝑆) → ((𝐸𝑦) = (𝐸𝑧) → 𝑦 = 𝑧)))
52 elsni 4414 . . . . . . . 8 (𝑦 ∈ {𝐾} → 𝑦 = 𝐾)
53 elsni 4414 . . . . . . . 8 (𝑧 ∈ {𝐾} → 𝑧 = 𝐾)
54 eqtr3 2848 . . . . . . . . 9 ((𝑦 = 𝐾𝑧 = 𝐾) → 𝑦 = 𝑧)
55542a1d 26 . . . . . . . 8 ((𝑦 = 𝐾𝑧 = 𝐾) → ((𝐾𝑁𝑍𝑆) → ((𝐸𝑦) = (𝐸𝑧) → 𝑦 = 𝑧)))
5652, 53, 55syl2an 591 . . . . . . 7 ((𝑦 ∈ {𝐾} ∧ 𝑧 ∈ {𝐾}) → ((𝐾𝑁𝑍𝑆) → ((𝐸𝑦) = (𝐸𝑧) → 𝑦 = 𝑧)))
5741, 47, 51, 56ccase 1066 . . . . . 6 (((𝑦 ∈ (𝑁 ∖ {𝐾}) ∨ 𝑦 ∈ {𝐾}) ∧ (𝑧 ∈ (𝑁 ∖ {𝐾}) ∨ 𝑧 ∈ {𝐾})) → ((𝐾𝑁𝑍𝑆) → ((𝐸𝑦) = (𝐸𝑧) → 𝑦 = 𝑧)))
5810, 11, 57syl2anb 593 . . . . 5 ((𝑦 ∈ ((𝑁 ∖ {𝐾}) ∪ {𝐾}) ∧ 𝑧 ∈ ((𝑁 ∖ {𝐾}) ∪ {𝐾})) → ((𝐾𝑁𝑍𝑆) → ((𝐸𝑦) = (𝐸𝑧) → 𝑦 = 𝑧)))
5958com12 32 . . . 4 ((𝐾𝑁𝑍𝑆) → ((𝑦 ∈ ((𝑁 ∖ {𝐾}) ∪ {𝐾}) ∧ 𝑧 ∈ ((𝑁 ∖ {𝐾}) ∪ {𝐾})) → ((𝐸𝑦) = (𝐸𝑧) → 𝑦 = 𝑧)))
609, 59sylbid 232 . . 3 ((𝐾𝑁𝑍𝑆) → ((𝑦𝑁𝑧𝑁) → ((𝐸𝑦) = (𝐸𝑧) → 𝑦 = 𝑧)))
6160ralrimivv 3179 . 2 ((𝐾𝑁𝑍𝑆) → ∀𝑦𝑁𝑧𝑁 ((𝐸𝑦) = (𝐸𝑧) → 𝑦 = 𝑧))
62 dff13 6767 . 2 (𝐸:𝑁1-1𝑁 ↔ (𝐸:𝑁𝑁 ∧ ∀𝑦𝑁𝑧𝑁 ((𝐸𝑦) = (𝐸𝑧) → 𝑦 = 𝑧)))
633, 61, 62sylanbrc 580 1 ((𝐾𝑁𝑍𝑆) → 𝐸:𝑁1-1𝑁)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  wo 880   = wceq 1658  wcel 2166  wne 2999  wral 3117  cdif 3795  cun 3796  ifcif 4306  {csn 4397  cmpt 4952  wf 6119  1-1wf1 6120  1-1-ontowf1o 6122  cfv 6123  Basecbs 16222  SymGrpcsymg 18147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209  ax-cnex 10308  ax-resscn 10309  ax-1cn 10310  ax-icn 10311  ax-addcl 10312  ax-addrcl 10313  ax-mulcl 10314  ax-mulrcl 10315  ax-mulcom 10316  ax-addass 10317  ax-mulass 10318  ax-distr 10319  ax-i2m1 10320  ax-1ne0 10321  ax-1rid 10322  ax-rnegex 10323  ax-rrecex 10324  ax-cnre 10325  ax-pre-lttri 10326  ax-pre-lttrn 10327  ax-pre-ltadd 10328  ax-pre-mulgt0 10329
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-int 4698  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-om 7327  df-1st 7428  df-2nd 7429  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-1o 7826  df-oadd 7830  df-er 8009  df-map 8124  df-en 8223  df-dom 8224  df-sdom 8225  df-fin 8226  df-pnf 10393  df-mnf 10394  df-xr 10395  df-ltxr 10396  df-le 10397  df-sub 10587  df-neg 10588  df-nn 11351  df-2 11414  df-3 11415  df-4 11416  df-5 11417  df-6 11418  df-7 11419  df-8 11420  df-9 11421  df-n0 11619  df-z 11705  df-uz 11969  df-fz 12620  df-struct 16224  df-ndx 16225  df-slot 16226  df-base 16228  df-plusg 16318  df-tset 16324  df-symg 18148
This theorem is referenced by:  symgextf1o  18193
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