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Theorem subfacp1lem4 30926
Description: Lemma for subfacp1 30929. The function 𝐹, which swaps 1 with 𝑀 and leaves all other elements alone, is a bijection of order 2, i.e. it is its own inverse. (Contributed by Mario Carneiro, 19-Jan-2015.)
Hypotheses
Ref Expression
derang.d 𝐷 = (𝑥 ∈ Fin ↦ (#‘{𝑓 ∣ (𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦)}))
subfac.n 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))
subfacp1lem.a 𝐴 = {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓𝑦) ≠ 𝑦)}
subfacp1lem1.n (𝜑𝑁 ∈ ℕ)
subfacp1lem1.m (𝜑𝑀 ∈ (2...(𝑁 + 1)))
subfacp1lem1.x 𝑀 ∈ V
subfacp1lem1.k 𝐾 = ((2...(𝑁 + 1)) ∖ {𝑀})
subfacp1lem5.b 𝐵 = {𝑔𝐴 ∣ ((𝑔‘1) = 𝑀 ∧ (𝑔𝑀) ≠ 1)}
subfacp1lem5.f 𝐹 = (( I ↾ 𝐾) ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})
Assertion
Ref Expression
subfacp1lem4 (𝜑𝐹 = 𝐹)
Distinct variable groups:   𝑓,𝑔,𝑛,𝑥,𝑦,𝐴   𝑓,𝐹,𝑔,𝑥,𝑦   𝑓,𝑁,𝑔,𝑛,𝑥,𝑦   𝐵,𝑓,𝑔,𝑥,𝑦   𝜑,𝑥,𝑦   𝐷,𝑛   𝑓,𝐾,𝑛,𝑥,𝑦   𝑓,𝑀,𝑔,𝑥,𝑦   𝑆,𝑛,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑓,𝑔,𝑛)   𝐵(𝑛)   𝐷(𝑥,𝑦,𝑓,𝑔)   𝑆(𝑓,𝑔)   𝐹(𝑛)   𝐾(𝑔)   𝑀(𝑛)

Proof of Theorem subfacp1lem4
StepHypRef Expression
1 derang.d . . . . 5 𝐷 = (𝑥 ∈ Fin ↦ (#‘{𝑓 ∣ (𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦)}))
2 subfac.n . . . . 5 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))
3 subfacp1lem.a . . . . 5 𝐴 = {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓𝑦) ≠ 𝑦)}
4 subfacp1lem1.n . . . . 5 (𝜑𝑁 ∈ ℕ)
5 subfacp1lem1.m . . . . 5 (𝜑𝑀 ∈ (2...(𝑁 + 1)))
6 subfacp1lem1.x . . . . 5 𝑀 ∈ V
7 subfacp1lem1.k . . . . 5 𝐾 = ((2...(𝑁 + 1)) ∖ {𝑀})
8 subfacp1lem5.f . . . . 5 𝐹 = (( I ↾ 𝐾) ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})
9 f1oi 6141 . . . . . 6 ( I ↾ 𝐾):𝐾1-1-onto𝐾
109a1i 11 . . . . 5 (𝜑 → ( I ↾ 𝐾):𝐾1-1-onto𝐾)
111, 2, 3, 4, 5, 6, 7, 8, 10subfacp1lem2a 30923 . . . 4 (𝜑 → (𝐹:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ (𝐹‘1) = 𝑀 ∧ (𝐹𝑀) = 1))
1211simp1d 1071 . . 3 (𝜑𝐹:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)))
13 f1ocnv 6116 . . 3 (𝐹:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) → 𝐹:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)))
14 f1ofn 6105 . . 3 (𝐹:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) → 𝐹 Fn (1...(𝑁 + 1)))
1512, 13, 143syl 18 . 2 (𝜑𝐹 Fn (1...(𝑁 + 1)))
16 f1ofn 6105 . . 3 (𝐹:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) → 𝐹 Fn (1...(𝑁 + 1)))
1712, 16syl 17 . 2 (𝜑𝐹 Fn (1...(𝑁 + 1)))
181, 2, 3, 4, 5, 6, 7subfacp1lem1 30922 . . . . . . . 8 (𝜑 → ((𝐾 ∩ {1, 𝑀}) = ∅ ∧ (𝐾 ∪ {1, 𝑀}) = (1...(𝑁 + 1)) ∧ (#‘𝐾) = (𝑁 − 1)))
1918simp2d 1072 . . . . . . 7 (𝜑 → (𝐾 ∪ {1, 𝑀}) = (1...(𝑁 + 1)))
2019eleq2d 2684 . . . . . 6 (𝜑 → (𝑥 ∈ (𝐾 ∪ {1, 𝑀}) ↔ 𝑥 ∈ (1...(𝑁 + 1))))
2120biimpar 502 . . . . 5 ((𝜑𝑥 ∈ (1...(𝑁 + 1))) → 𝑥 ∈ (𝐾 ∪ {1, 𝑀}))
22 elun 3737 . . . . 5 (𝑥 ∈ (𝐾 ∪ {1, 𝑀}) ↔ (𝑥𝐾𝑥 ∈ {1, 𝑀}))
2321, 22sylib 208 . . . 4 ((𝜑𝑥 ∈ (1...(𝑁 + 1))) → (𝑥𝐾𝑥 ∈ {1, 𝑀}))
241, 2, 3, 4, 5, 6, 7, 8, 10subfacp1lem2b 30924 . . . . . . . 8 ((𝜑𝑥𝐾) → (𝐹𝑥) = (( I ↾ 𝐾)‘𝑥))
25 fvresi 6404 . . . . . . . . 9 (𝑥𝐾 → (( I ↾ 𝐾)‘𝑥) = 𝑥)
2625adantl 482 . . . . . . . 8 ((𝜑𝑥𝐾) → (( I ↾ 𝐾)‘𝑥) = 𝑥)
2724, 26eqtrd 2655 . . . . . . 7 ((𝜑𝑥𝐾) → (𝐹𝑥) = 𝑥)
2827fveq2d 6162 . . . . . 6 ((𝜑𝑥𝐾) → (𝐹‘(𝐹𝑥)) = (𝐹𝑥))
2928, 27eqtrd 2655 . . . . 5 ((𝜑𝑥𝐾) → (𝐹‘(𝐹𝑥)) = 𝑥)
30 vex 3193 . . . . . . 7 𝑥 ∈ V
3130elpr 4176 . . . . . 6 (𝑥 ∈ {1, 𝑀} ↔ (𝑥 = 1 ∨ 𝑥 = 𝑀))
3211simp2d 1072 . . . . . . . . . . 11 (𝜑 → (𝐹‘1) = 𝑀)
3332fveq2d 6162 . . . . . . . . . 10 (𝜑 → (𝐹‘(𝐹‘1)) = (𝐹𝑀))
3411simp3d 1073 . . . . . . . . . 10 (𝜑 → (𝐹𝑀) = 1)
3533, 34eqtrd 2655 . . . . . . . . 9 (𝜑 → (𝐹‘(𝐹‘1)) = 1)
36 fveq2 6158 . . . . . . . . . . 11 (𝑥 = 1 → (𝐹𝑥) = (𝐹‘1))
3736fveq2d 6162 . . . . . . . . . 10 (𝑥 = 1 → (𝐹‘(𝐹𝑥)) = (𝐹‘(𝐹‘1)))
38 id 22 . . . . . . . . . 10 (𝑥 = 1 → 𝑥 = 1)
3937, 38eqeq12d 2636 . . . . . . . . 9 (𝑥 = 1 → ((𝐹‘(𝐹𝑥)) = 𝑥 ↔ (𝐹‘(𝐹‘1)) = 1))
4035, 39syl5ibrcom 237 . . . . . . . 8 (𝜑 → (𝑥 = 1 → (𝐹‘(𝐹𝑥)) = 𝑥))
4134fveq2d 6162 . . . . . . . . . 10 (𝜑 → (𝐹‘(𝐹𝑀)) = (𝐹‘1))
4241, 32eqtrd 2655 . . . . . . . . 9 (𝜑 → (𝐹‘(𝐹𝑀)) = 𝑀)
43 fveq2 6158 . . . . . . . . . . 11 (𝑥 = 𝑀 → (𝐹𝑥) = (𝐹𝑀))
4443fveq2d 6162 . . . . . . . . . 10 (𝑥 = 𝑀 → (𝐹‘(𝐹𝑥)) = (𝐹‘(𝐹𝑀)))
45 id 22 . . . . . . . . . 10 (𝑥 = 𝑀𝑥 = 𝑀)
4644, 45eqeq12d 2636 . . . . . . . . 9 (𝑥 = 𝑀 → ((𝐹‘(𝐹𝑥)) = 𝑥 ↔ (𝐹‘(𝐹𝑀)) = 𝑀))
4742, 46syl5ibrcom 237 . . . . . . . 8 (𝜑 → (𝑥 = 𝑀 → (𝐹‘(𝐹𝑥)) = 𝑥))
4840, 47jaod 395 . . . . . . 7 (𝜑 → ((𝑥 = 1 ∨ 𝑥 = 𝑀) → (𝐹‘(𝐹𝑥)) = 𝑥))
4948imp 445 . . . . . 6 ((𝜑 ∧ (𝑥 = 1 ∨ 𝑥 = 𝑀)) → (𝐹‘(𝐹𝑥)) = 𝑥)
5031, 49sylan2b 492 . . . . 5 ((𝜑𝑥 ∈ {1, 𝑀}) → (𝐹‘(𝐹𝑥)) = 𝑥)
5129, 50jaodan 825 . . . 4 ((𝜑 ∧ (𝑥𝐾𝑥 ∈ {1, 𝑀})) → (𝐹‘(𝐹𝑥)) = 𝑥)
5223, 51syldan 487 . . 3 ((𝜑𝑥 ∈ (1...(𝑁 + 1))) → (𝐹‘(𝐹𝑥)) = 𝑥)
5312adantr 481 . . . 4 ((𝜑𝑥 ∈ (1...(𝑁 + 1))) → 𝐹:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)))
54 f1of 6104 . . . . . 6 (𝐹:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) → 𝐹:(1...(𝑁 + 1))⟶(1...(𝑁 + 1)))
5512, 54syl 17 . . . . 5 (𝜑𝐹:(1...(𝑁 + 1))⟶(1...(𝑁 + 1)))
5655ffvelrnda 6325 . . . 4 ((𝜑𝑥 ∈ (1...(𝑁 + 1))) → (𝐹𝑥) ∈ (1...(𝑁 + 1)))
57 f1ocnvfv 6499 . . . 4 ((𝐹:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ (𝐹𝑥) ∈ (1...(𝑁 + 1))) → ((𝐹‘(𝐹𝑥)) = 𝑥 → (𝐹𝑥) = (𝐹𝑥)))
5853, 56, 57syl2anc 692 . . 3 ((𝜑𝑥 ∈ (1...(𝑁 + 1))) → ((𝐹‘(𝐹𝑥)) = 𝑥 → (𝐹𝑥) = (𝐹𝑥)))
5952, 58mpd 15 . 2 ((𝜑𝑥 ∈ (1...(𝑁 + 1))) → (𝐹𝑥) = (𝐹𝑥))
6015, 17, 59eqfnfvd 6280 1 (𝜑𝐹 = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 383  wa 384   = wceq 1480  wcel 1987  {cab 2607  wne 2790  wral 2908  {crab 2912  Vcvv 3190  cdif 3557  cun 3558  cin 3559  c0 3897  {csn 4155  {cpr 4157  cop 4161  cmpt 4683   I cid 4994  ccnv 5083  cres 5086   Fn wfn 5852  wf 5853  1-1-ontowf1o 5856  cfv 5857  (class class class)co 6615  Fincfn 7915  1c1 9897   + caddc 9899  cmin 10226  cn 10980  2c2 11030  0cn0 11252  ...cfz 12284  #chash 13073
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-oadd 7524  df-er 7702  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-card 8725  df-cda 8950  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-nn 10981  df-2 11039  df-n0 11253  df-z 11338  df-uz 11648  df-fz 12285  df-hash 13074
This theorem is referenced by:  subfacp1lem5  30927
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