Theorem subfacp1lem4 32432

Description: Lemma for subfacp1 32435. 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 6654	. . . . . 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 32429	. . . 4 ⊢ (𝜑 → (𝐹:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ (𝐹‘1) = 𝑀 ∧ (𝐹‘𝑀) = 1))
12	11	simp1d 1138	. . 3 ⊢ (𝜑 → 𝐹:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)))
13		f1ocnv 6629	. . 3 ⊢ (𝐹:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) → ◡𝐹:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)))
14		f1ofn 6618	. . 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 6618	. . 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 32428	. . . . . . . 8 ⊢ (𝜑 → ((𝐾 ∩ {1, 𝑀}) = ∅ ∧ (𝐾 ∪ {1, 𝑀}) = (1...(𝑁 + 1)) ∧ (♯‘𝐾) = (𝑁 − 1)))
19	18	simp2d 1139	. . . . . . 7 ⊢ (𝜑 → (𝐾 ∪ {1, 𝑀}) = (1...(𝑁 + 1)))
20	19	eleq2d 2900	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝐾 ∪ {1, 𝑀}) ↔ 𝑥 ∈ (1...(𝑁 + 1))))
21	20	biimpar 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(𝑁 + 1))) → 𝑥 ∈ (𝐾 ∪ {1, 𝑀}))
22		elun 4127	. . . . 5 ⊢ (𝑥 ∈ (𝐾 ∪ {1, 𝑀}) ↔ (𝑥 ∈ 𝐾 ∨ 𝑥 ∈ {1, 𝑀}))
23	21, 22	sylib 220	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(𝑁 + 1))) → (𝑥 ∈ 𝐾 ∨ 𝑥 ∈ {1, 𝑀}))
24	1, 2, 3, 4, 5, 6, 7, 8, 10	subfacp1lem2b 32430	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → (𝐹‘𝑥) = (( I ↾ 𝐾)‘𝑥))
25		fvresi 6937	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐾 → (( I ↾ 𝐾)‘𝑥) = 𝑥)
26	25	adantl 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → (( I ↾ 𝐾)‘𝑥) = 𝑥)
27	24, 26	eqtrd 2858	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → (𝐹‘𝑥) = 𝑥)
28	27	fveq2d 6676	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥))
29	28, 27	eqtrd 2858	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → (𝐹‘(𝐹‘𝑥)) = 𝑥)
30		vex 3499	. . . . . . 7 ⊢ 𝑥 ∈ V
31	30	elpr 4592	. . . . . 6 ⊢ (𝑥 ∈ {1, 𝑀} ↔ (𝑥 = 1 ∨ 𝑥 = 𝑀))
32	11	simp2d 1139	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹‘1) = 𝑀)
33	32	fveq2d 6676	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹‘(𝐹‘1)) = (𝐹‘𝑀))
34	11	simp3d 1140	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹‘𝑀) = 1)
35	33, 34	eqtrd 2858	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘(𝐹‘1)) = 1)
36		2fveq3 6677	. . . . . . . . . 10 ⊢ (𝑥 = 1 → (𝐹‘(𝐹‘𝑥)) = (𝐹‘(𝐹‘1)))
37		id 22	. . . . . . . . . 10 ⊢ (𝑥 = 1 → 𝑥 = 1)
38	36, 37	eqeq12d 2839	. . . . . . . . 9 ⊢ (𝑥 = 1 → ((𝐹‘(𝐹‘𝑥)) = 𝑥 ↔ (𝐹‘(𝐹‘1)) = 1))
39	35, 38	syl5ibrcom 249	. . . . . . . 8 ⊢ (𝜑 → (𝑥 = 1 → (𝐹‘(𝐹‘𝑥)) = 𝑥))
40	34	fveq2d 6676	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹‘(𝐹‘𝑀)) = (𝐹‘1))
41	40, 32	eqtrd 2858	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘(𝐹‘𝑀)) = 𝑀)
42		2fveq3 6677	. . . . . . . . . 10 ⊢ (𝑥 = 𝑀 → (𝐹‘(𝐹‘𝑥)) = (𝐹‘(𝐹‘𝑀)))
43		id 22	. . . . . . . . . 10 ⊢ (𝑥 = 𝑀 → 𝑥 = 𝑀)
44	42, 43	eqeq12d 2839	. . . . . . . . 9 ⊢ (𝑥 = 𝑀 → ((𝐹‘(𝐹‘𝑥)) = 𝑥 ↔ (𝐹‘(𝐹‘𝑀)) = 𝑀))
45	41, 44	syl5ibrcom 249	. . . . . . . 8 ⊢ (𝜑 → (𝑥 = 𝑀 → (𝐹‘(𝐹‘𝑥)) = 𝑥))
46	39, 45	jaod 855	. . . . . . 7 ⊢ (𝜑 → ((𝑥 = 1 ∨ 𝑥 = 𝑀) → (𝐹‘(𝐹‘𝑥)) = 𝑥))
47	46	imp 409	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = 1 ∨ 𝑥 = 𝑀)) → (𝐹‘(𝐹‘𝑥)) = 𝑥)
48	31, 47	sylan2b 595	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ {1, 𝑀}) → (𝐹‘(𝐹‘𝑥)) = 𝑥)
49	29, 48	jaodan 954	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∨ 𝑥 ∈ {1, 𝑀})) → (𝐹‘(𝐹‘𝑥)) = 𝑥)
50	23, 49	syldan 593	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(𝑁 + 1))) → (𝐹‘(𝐹‘𝑥)) = 𝑥)
51	12	adantr 483	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(𝑁 + 1))) → 𝐹:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)))
52		f1of 6617	. . . . . 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 6853	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(𝑁 + 1))) → (𝐹‘𝑥) ∈ (1...(𝑁 + 1)))
55		f1ocnvfv 7037	. . . 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 586	. . 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 6807	1 ⊢ (𝜑 → ◡𝐹 = 𝐹)

Syntax hints:   → wi 4   ∧ wa 398   ∨ wo 843   = wceq 1537   ∈ wcel 2114  {cab 2801   ≠ wne 3018  ∀wral 3140  {crab 3144  Vcvv 3496   ∖ cdif 3935   ∪ cun 3936   ∩ cin 3937  ∅c0 4293  {csn 4569  {cpr 4571  ⟨cop 4575   ↦ cmpt 5148   I cid 5461  ^◡ccnv 5556   ↾ cres 5559   Fn wfn 6352  ⟶wf 6353  –1-1-onto→wf1o 6356  ‘cfv 6357  (class class class)co 7158  Fincfn 8511  1c1 10540   + caddc 10542   − cmin 10872  ℕcn 11640  2c2 11695  ℕ₀cn0 11900  ...cfz 12895  ♯chash 13693

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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-1st 7691  df-2nd 7692  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-1o 8104  df-oadd 8108  df-er 8291  df-en 8512  df-dom 8513  df-sdom 8514  df-fin 8515  df-dju 9332  df-card 9370  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-nn 11641  df-2 11703  df-n0 11901  df-z 11985  df-uz 12247  df-fz 12896  df-hash 13694

This theorem is referenced by:  subfacp1lem5  32433