Theorem subfacp1lem4 33145

Description: Lemma for subfacp1 33148. 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:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 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))(𝑓‘𝑦) ≠ 𝑦)}
subfacp1lem1.n	⊢ (𝜑 → 𝑁 ∈ ℕ)
subfacp1lem1.m	⊢ (𝜑 → 𝑀 ∈ (2...(𝑁 + 1)))
subfacp1lem1.x	⊢ 𝑀 ∈ V
subfacp1lem1.k	⊢ 𝐾 = ((2...(𝑁 + 1)) ∖ {𝑀})
subfacp1lem5.b	⊢ 𝐵 = {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = 𝑀 ∧ (𝑔‘𝑀) ≠ 1)}
subfacp1lem5.f	⊢ 𝐹 = (( I ↾ 𝐾) ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})

Assertion

Ref	Expression
subfacp1lem4	⊢ (𝜑 → ◡𝐹 = 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

Step	Hyp	Ref	Expression
1		derang.d	. . . . 5 ⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 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))(𝑓‘𝑦) ≠ 𝑦)}
4		subfacp1lem1.n	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ)
5		subfacp1lem1.m	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ (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 6754	. . . . . 6 ⊢ ( I ↾ 𝐾):𝐾–1-1-onto→𝐾
10	9	a1i 11	. . . . 5 ⊢ (𝜑 → ( I ↾ 𝐾):𝐾–1-1-onto→𝐾)
11	1, 2, 3, 4, 5, 6, 7, 8, 10	subfacp1lem2a 33142	. . . 4 ⊢ (𝜑 → (𝐹:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ (𝐹‘1) = 𝑀 ∧ (𝐹‘𝑀) = 1))
12	11	simp1d 1141	. . 3 ⊢ (𝜑 → 𝐹:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)))
13		f1ocnv 6728	. . 3 ⊢ (𝐹:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) → ◡𝐹:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)))
14		f1ofn 6717	. . 3 ⊢ (◡𝐹:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) → ◡𝐹 Fn (1...(𝑁 + 1)))
15	12, 13, 14	3syl 18	. 2 ⊢ (𝜑 → ◡𝐹 Fn (1...(𝑁 + 1)))
16		f1ofn 6717	. . 3 ⊢ (𝐹:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) → 𝐹 Fn (1...(𝑁 + 1)))
17	12, 16	syl 17	. 2 ⊢ (𝜑 → 𝐹 Fn (1...(𝑁 + 1)))
18	1, 2, 3, 4, 5, 6, 7	subfacp1lem1 33141	. . . . . . . 8 ⊢ (𝜑 → ((𝐾 ∩ {1, 𝑀}) = ∅ ∧ (𝐾 ∪ {1, 𝑀}) = (1...(𝑁 + 1)) ∧ (♯‘𝐾) = (𝑁 − 1)))
19	18	simp2d 1142	. . . . . . 7 ⊢ (𝜑 → (𝐾 ∪ {1, 𝑀}) = (1...(𝑁 + 1)))
20	19	eleq2d 2824	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝐾 ∪ {1, 𝑀}) ↔ 𝑥 ∈ (1...(𝑁 + 1))))
21	20	biimpar 478	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(𝑁 + 1))) → 𝑥 ∈ (𝐾 ∪ {1, 𝑀}))
22		elun 4083	. . . . 5 ⊢ (𝑥 ∈ (𝐾 ∪ {1, 𝑀}) ↔ (𝑥 ∈ 𝐾 ∨ 𝑥 ∈ {1, 𝑀}))
23	21, 22	sylib 217	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(𝑁 + 1))) → (𝑥 ∈ 𝐾 ∨ 𝑥 ∈ {1, 𝑀}))
24	1, 2, 3, 4, 5, 6, 7, 8, 10	subfacp1lem2b 33143	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → (𝐹‘𝑥) = (( I ↾ 𝐾)‘𝑥))
25		fvresi 7045	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐾 → (( I ↾ 𝐾)‘𝑥) = 𝑥)
26	25	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → (( I ↾ 𝐾)‘𝑥) = 𝑥)
27	24, 26	eqtrd 2778	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → (𝐹‘𝑥) = 𝑥)
28	27	fveq2d 6778	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥))
29	28, 27	eqtrd 2778	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → (𝐹‘(𝐹‘𝑥)) = 𝑥)
30		vex 3436	. . . . . . 7 ⊢ 𝑥 ∈ V
31	30	elpr 4584	. . . . . 6 ⊢ (𝑥 ∈ {1, 𝑀} ↔ (𝑥 = 1 ∨ 𝑥 = 𝑀))
32	11	simp2d 1142	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹‘1) = 𝑀)
33	32	fveq2d 6778	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹‘(𝐹‘1)) = (𝐹‘𝑀))
34	11	simp3d 1143	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹‘𝑀) = 1)
35	33, 34	eqtrd 2778	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘(𝐹‘1)) = 1)
36		2fveq3 6779	. . . . . . . . . 10 ⊢ (𝑥 = 1 → (𝐹‘(𝐹‘𝑥)) = (𝐹‘(𝐹‘1)))
37		id 22	. . . . . . . . . 10 ⊢ (𝑥 = 1 → 𝑥 = 1)
38	36, 37	eqeq12d 2754	. . . . . . . . 9 ⊢ (𝑥 = 1 → ((𝐹‘(𝐹‘𝑥)) = 𝑥 ↔ (𝐹‘(𝐹‘1)) = 1))
39	35, 38	syl5ibrcom 246	. . . . . . . 8 ⊢ (𝜑 → (𝑥 = 1 → (𝐹‘(𝐹‘𝑥)) = 𝑥))
40	34	fveq2d 6778	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹‘(𝐹‘𝑀)) = (𝐹‘1))
41	40, 32	eqtrd 2778	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘(𝐹‘𝑀)) = 𝑀)
42		2fveq3 6779	. . . . . . . . . 10 ⊢ (𝑥 = 𝑀 → (𝐹‘(𝐹‘𝑥)) = (𝐹‘(𝐹‘𝑀)))
43		id 22	. . . . . . . . . 10 ⊢ (𝑥 = 𝑀 → 𝑥 = 𝑀)
44	42, 43	eqeq12d 2754	. . . . . . . . 9 ⊢ (𝑥 = 𝑀 → ((𝐹‘(𝐹‘𝑥)) = 𝑥 ↔ (𝐹‘(𝐹‘𝑀)) = 𝑀))
45	41, 44	syl5ibrcom 246	. . . . . . . 8 ⊢ (𝜑 → (𝑥 = 𝑀 → (𝐹‘(𝐹‘𝑥)) = 𝑥))
46	39, 45	jaod 856	. . . . . . 7 ⊢ (𝜑 → ((𝑥 = 1 ∨ 𝑥 = 𝑀) → (𝐹‘(𝐹‘𝑥)) = 𝑥))
47	46	imp 407	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = 1 ∨ 𝑥 = 𝑀)) → (𝐹‘(𝐹‘𝑥)) = 𝑥)
48	31, 47	sylan2b 594	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ {1, 𝑀}) → (𝐹‘(𝐹‘𝑥)) = 𝑥)
49	29, 48	jaodan 955	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∨ 𝑥 ∈ {1, 𝑀})) → (𝐹‘(𝐹‘𝑥)) = 𝑥)
50	23, 49	syldan 591	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(𝑁 + 1))) → (𝐹‘(𝐹‘𝑥)) = 𝑥)
51	12	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(𝑁 + 1))) → 𝐹:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)))
52		f1of 6716	. . . . . 6 ⊢ (𝐹:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) → 𝐹:(1...(𝑁 + 1))⟶(1...(𝑁 + 1)))
53	12, 52	syl 17	. . . . 5 ⊢ (𝜑 → 𝐹:(1...(𝑁 + 1))⟶(1...(𝑁 + 1)))
54	53	ffvelrnda 6961	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(𝑁 + 1))) → (𝐹‘𝑥) ∈ (1...(𝑁 + 1)))
55		f1ocnvfv 7150	. . . 4 ⊢ ((𝐹:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ (𝐹‘𝑥) ∈ (1...(𝑁 + 1))) → ((𝐹‘(𝐹‘𝑥)) = 𝑥 → (◡𝐹‘𝑥) = (𝐹‘𝑥)))
56	51, 54, 55	syl2anc 584	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(𝑁 + 1))) → ((𝐹‘(𝐹‘𝑥)) = 𝑥 → (◡𝐹‘𝑥) = (𝐹‘𝑥)))
57	50, 56	mpd 15	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(𝑁 + 1))) → (◡𝐹‘𝑥) = (𝐹‘𝑥))
58	15, 17, 57	eqfnfvd 6912	1 ⊢ (𝜑 → ◡𝐹 = 𝐹)

Syntax hints:   → wi 4   ∧ wa 396   ∨ wo 844   = wceq 1539   ∈ wcel 2106  {cab 2715   ≠ wne 2943  ∀wral 3064  {crab 3068  Vcvv 3432   ∖ cdif 3884   ∪ cun 3885   ∩ cin 3886  ∅c0 4256  {csn 4561  {cpr 4563  ⟨cop 4567   ↦ cmpt 5157   I cid 5488  ^◡ccnv 5588   ↾ cres 5591   Fn wfn 6428  ⟶wf 6429  –1-1-onto→wf1o 6432  ‘cfv 6433  (class class class)co 7275  Fincfn 8733  1c1 10872   + caddc 10874   − cmin 11205  ℕcn 11973  2c2 12028  ℕ₀cn0 12233  ...cfz 13239  ♯chash 14044

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-oadd 8301  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-dju 9659  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-n0 12234  df-z 12320  df-uz 12583  df-fz 13240  df-hash 14045

This theorem is referenced by:  subfacp1lem5  33146