Theorem nvocnv 7256

Description: The converse of an involution is the function itself. (Contributed by Thierry Arnoux, 7-May-2019.)

Assertion

Ref	Expression
nvocnv	⊢ ((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) → ◡𝐹 = 𝐹)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐹

Proof of Theorem nvocnv

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simprr 772	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) ∧ (𝑧 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑧))) → 𝑦 = (𝐹‘𝑧))
2		simpll 766	. . . . . . 7 ⊢ (((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) ∧ (𝑧 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑧))) → 𝐹:𝐴⟶𝐴)
3		simprl 770	. . . . . . 7 ⊢ (((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) ∧ (𝑧 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑧))) → 𝑧 ∈ 𝐴)
4	2, 3	ffvelcdmd 7057	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) ∧ (𝑧 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑧))) → (𝐹‘𝑧) ∈ 𝐴)
5	1, 4	eqeltrd 2828	. . . . 5 ⊢ (((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) ∧ (𝑧 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑧))) → 𝑦 ∈ 𝐴)
6	1	fveq2d 6862	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) ∧ (𝑧 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑧))) → (𝐹‘𝑦) = (𝐹‘(𝐹‘𝑧)))
7		2fveq3 6863	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝐹‘(𝐹‘𝑥)) = (𝐹‘(𝐹‘𝑧)))
8		id 22	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → 𝑥 = 𝑧)
9	7, 8	eqeq12d 2745	. . . . . . 7 ⊢ (𝑥 = 𝑧 → ((𝐹‘(𝐹‘𝑥)) = 𝑥 ↔ (𝐹‘(𝐹‘𝑧)) = 𝑧))
10		simplr 768	. . . . . . 7 ⊢ (((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) ∧ (𝑧 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑧))) → ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥)
11	9, 10, 3	rspcdva 3589	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) ∧ (𝑧 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑧))) → (𝐹‘(𝐹‘𝑧)) = 𝑧)
12	6, 11	eqtr2d 2765	. . . . 5 ⊢ (((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) ∧ (𝑧 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑧))) → 𝑧 = (𝐹‘𝑦))
13	5, 12	jca 511	. . . 4 ⊢ (((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) ∧ (𝑧 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑧))) → (𝑦 ∈ 𝐴 ∧ 𝑧 = (𝐹‘𝑦)))
14		simprr 772	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = (𝐹‘𝑦))) → 𝑧 = (𝐹‘𝑦))
15		simpll 766	. . . . . . 7 ⊢ (((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = (𝐹‘𝑦))) → 𝐹:𝐴⟶𝐴)
16		simprl 770	. . . . . . 7 ⊢ (((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = (𝐹‘𝑦))) → 𝑦 ∈ 𝐴)
17	15, 16	ffvelcdmd 7057	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = (𝐹‘𝑦))) → (𝐹‘𝑦) ∈ 𝐴)
18	14, 17	eqeltrd 2828	. . . . 5 ⊢ (((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = (𝐹‘𝑦))) → 𝑧 ∈ 𝐴)
19	14	fveq2d 6862	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = (𝐹‘𝑦))) → (𝐹‘𝑧) = (𝐹‘(𝐹‘𝑦)))
20		2fveq3 6863	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐹‘(𝐹‘𝑥)) = (𝐹‘(𝐹‘𝑦)))
21		id 22	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
22	20, 21	eqeq12d 2745	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝐹‘(𝐹‘𝑥)) = 𝑥 ↔ (𝐹‘(𝐹‘𝑦)) = 𝑦))
23		simplr 768	. . . . . . 7 ⊢ (((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = (𝐹‘𝑦))) → ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥)
24	22, 23, 16	rspcdva 3589	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = (𝐹‘𝑦))) → (𝐹‘(𝐹‘𝑦)) = 𝑦)
25	19, 24	eqtr2d 2765	. . . . 5 ⊢ (((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = (𝐹‘𝑦))) → 𝑦 = (𝐹‘𝑧))
26	18, 25	jca 511	. . . 4 ⊢ (((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = (𝐹‘𝑦))) → (𝑧 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑧)))
27	13, 26	impbida 800	. . 3 ⊢ ((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) → ((𝑧 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑧)) ↔ (𝑦 ∈ 𝐴 ∧ 𝑧 = (𝐹‘𝑦))))
28	27	mptcnv 6112	. 2 ⊢ ((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) → ◡(𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧)) = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)))
29		ffn 6688	. . . 4 ⊢ (𝐹:𝐴⟶𝐴 → 𝐹 Fn 𝐴)
30		dffn5 6919	. . . . . 6 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧)))
31	30	biimpi 216	. . . . 5 ⊢ (𝐹 Fn 𝐴 → 𝐹 = (𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧)))
32	31	adantr 480	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) → 𝐹 = (𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧)))
33	29, 32	sylan 580	. . 3 ⊢ ((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) → 𝐹 = (𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧)))
34	33	cnveqd 5839	. 2 ⊢ ((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) → ◡𝐹 = ◡(𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧)))
35		dffn5 6919	. . . . 5 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)))
36	35	biimpi 216	. . . 4 ⊢ (𝐹 Fn 𝐴 → 𝐹 = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)))
37	36	adantr 480	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) → 𝐹 = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)))
38	29, 37	sylan 580	. 2 ⊢ ((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) → 𝐹 = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)))
39	28, 34, 38	3eqtr4d 2774	1 ⊢ ((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) → ◡𝐹 = 𝐹)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044   ↦ cmpt 5188  ^◡ccnv 5637   Fn wfn 6506  ⟶wf 6507  ‘cfv 6511

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-fv 6519

This theorem is referenced by:  mirf1o  28596  lmif1o  28722  nvocnvb  43411