Theorem nvocnv 6437

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 792	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) ∧ (𝑧 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑧))) → 𝑦 = (𝐹‘𝑧))
2		simpll 786	. . . . . . 7 ⊢ (((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) ∧ (𝑧 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑧))) → 𝐹:𝐴⟶𝐴)
3		simprl 790	. . . . . . 7 ⊢ (((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) ∧ (𝑧 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑧))) → 𝑧 ∈ 𝐴)
4	2, 3	ffvelrnd 6268	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) ∧ (𝑧 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑧))) → (𝐹‘𝑧) ∈ 𝐴)
5	1, 4	eqeltrd 2688	. . . . 5 ⊢ (((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) ∧ (𝑧 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑧))) → 𝑦 ∈ 𝐴)
6	1	fveq2d 6107	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) ∧ (𝑧 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑧))) → (𝐹‘𝑦) = (𝐹‘(𝐹‘𝑧)))
7		simplr 788	. . . . . . 7 ⊢ (((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) ∧ (𝑧 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑧))) → ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥)
8		fveq2 6103	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
9	8	fveq2d 6107	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝐹‘(𝐹‘𝑥)) = (𝐹‘(𝐹‘𝑧)))
10		id 22	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → 𝑥 = 𝑧)
11	9, 10	eqeq12d 2625	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → ((𝐹‘(𝐹‘𝑥)) = 𝑥 ↔ (𝐹‘(𝐹‘𝑧)) = 𝑧))
12	11	rspcv 3278	. . . . . . 7 ⊢ (𝑧 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥 → (𝐹‘(𝐹‘𝑧)) = 𝑧))
13	3, 7, 12	sylc 63	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) ∧ (𝑧 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑧))) → (𝐹‘(𝐹‘𝑧)) = 𝑧)
14	6, 13	eqtr2d 2645	. . . . 5 ⊢ (((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) ∧ (𝑧 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑧))) → 𝑧 = (𝐹‘𝑦))
15	5, 14	jca 553	. . . 4 ⊢ (((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) ∧ (𝑧 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑧))) → (𝑦 ∈ 𝐴 ∧ 𝑧 = (𝐹‘𝑦)))
16		simprr 792	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = (𝐹‘𝑦))) → 𝑧 = (𝐹‘𝑦))
17		simpll 786	. . . . . . 7 ⊢ (((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = (𝐹‘𝑦))) → 𝐹:𝐴⟶𝐴)
18		simprl 790	. . . . . . 7 ⊢ (((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = (𝐹‘𝑦))) → 𝑦 ∈ 𝐴)
19	17, 18	ffvelrnd 6268	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = (𝐹‘𝑦))) → (𝐹‘𝑦) ∈ 𝐴)
20	16, 19	eqeltrd 2688	. . . . 5 ⊢ (((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = (𝐹‘𝑦))) → 𝑧 ∈ 𝐴)
21	16	fveq2d 6107	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = (𝐹‘𝑦))) → (𝐹‘𝑧) = (𝐹‘(𝐹‘𝑦)))
22		simplr 788	. . . . . . 7 ⊢ (((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = (𝐹‘𝑦))) → ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥)
23		fveq2 6103	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
24	23	fveq2d 6107	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝐹‘(𝐹‘𝑥)) = (𝐹‘(𝐹‘𝑦)))
25		id 22	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
26	24, 25	eqeq12d 2625	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝐹‘(𝐹‘𝑥)) = 𝑥 ↔ (𝐹‘(𝐹‘𝑦)) = 𝑦))
27	26	rspcv 3278	. . . . . . 7 ⊢ (𝑦 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥 → (𝐹‘(𝐹‘𝑦)) = 𝑦))
28	18, 22, 27	sylc 63	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = (𝐹‘𝑦))) → (𝐹‘(𝐹‘𝑦)) = 𝑦)
29	21, 28	eqtr2d 2645	. . . . 5 ⊢ (((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = (𝐹‘𝑦))) → 𝑦 = (𝐹‘𝑧))
30	20, 29	jca 553	. . . 4 ⊢ (((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = (𝐹‘𝑦))) → (𝑧 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑧)))
31	15, 30	impbida 873	. . 3 ⊢ ((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) → ((𝑧 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑧)) ↔ (𝑦 ∈ 𝐴 ∧ 𝑧 = (𝐹‘𝑦))))
32	31	mptcnv 5453	. 2 ⊢ ((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) → ◡(𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧)) = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)))
33		ffn 5958	. . . 4 ⊢ (𝐹:𝐴⟶𝐴 → 𝐹 Fn 𝐴)
34		dffn5 6151	. . . . . 6 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧)))
35	34	biimpi 205	. . . . 5 ⊢ (𝐹 Fn 𝐴 → 𝐹 = (𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧)))
36	35	adantr 480	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) → 𝐹 = (𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧)))
37	33, 36	sylan 487	. . 3 ⊢ ((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) → 𝐹 = (𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧)))
38	37	cnveqd 5220	. 2 ⊢ ((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) → ◡𝐹 = ◡(𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧)))
39		dffn5 6151	. . . . 5 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)))
40	39	biimpi 205	. . . 4 ⊢ (𝐹 Fn 𝐴 → 𝐹 = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)))
41	40	adantr 480	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) → 𝐹 = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)))
42	33, 41	sylan 487	. 2 ⊢ ((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) → 𝐹 = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)))
43	32, 38, 42	3eqtr4d 2654	1 ⊢ ((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) → ◡𝐹 = 𝐹)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ↦ cmpt 4643  ^◡ccnv 5037   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812

This theorem is referenced by:  mirf1o  25364  lmif1o  25487