Theorem nvocnv 7267

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 782	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) ∧ (𝑧 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑧))) → 𝑦 = (𝐹‘𝑧))
2		simpll 776	. . . . . . 7 ⊢ (((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) ∧ (𝑧 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑧))) → 𝐹:𝐴⟶𝐴)
3		simprl 780	. . . . . . 7 ⊢ (((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) ∧ (𝑧 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑧))) → 𝑧 ∈ 𝐴)
4	2, 3	ffvelcdmd 7068	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) ∧ (𝑧 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑧))) → (𝐹‘𝑧) ∈ 𝐴)
5	1, 4	eqeltrd 2864	. . . . 5 ⊢ (((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) ∧ (𝑧 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑧))) → 𝑦 ∈ 𝐴)
6	1	fveq2d 6873	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) ∧ (𝑧 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑧))) → (𝐹‘𝑦) = (𝐹‘(𝐹‘𝑧)))
7		2fveq3 6874	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝐹‘(𝐹‘𝑥)) = (𝐹‘(𝐹‘𝑧)))
8		id 22	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → 𝑥 = 𝑧)
9	7, 8	eqeq12d 2780	. . . . . . 7 ⊢ (𝑥 = 𝑧 → ((𝐹‘(𝐹‘𝑥)) = 𝑥 ↔ (𝐹‘(𝐹‘𝑧)) = 𝑧))
10		simplr 778	. . . . . . 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 3584	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) ∧ (𝑧 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑧))) → (𝐹‘(𝐹‘𝑧)) = 𝑧)
12	6, 11	eqtr2d 2800	. . . . 5 ⊢ (((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) ∧ (𝑧 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑧))) → 𝑧 = (𝐹‘𝑦))
13	5, 12	jca 519	. . . 4 ⊢ (((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) ∧ (𝑧 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑧))) → (𝑦 ∈ 𝐴 ∧ 𝑧 = (𝐹‘𝑦)))
14		simprr 782	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = (𝐹‘𝑦))) → 𝑧 = (𝐹‘𝑦))
15		simpll 776	. . . . . . 7 ⊢ (((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = (𝐹‘𝑦))) → 𝐹:𝐴⟶𝐴)
16		simprl 780	. . . . . . 7 ⊢ (((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = (𝐹‘𝑦))) → 𝑦 ∈ 𝐴)
17	15, 16	ffvelcdmd 7068	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = (𝐹‘𝑦))) → (𝐹‘𝑦) ∈ 𝐴)
18	14, 17	eqeltrd 2864	. . . . 5 ⊢ (((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = (𝐹‘𝑦))) → 𝑧 ∈ 𝐴)
19	14	fveq2d 6873	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = (𝐹‘𝑦))) → (𝐹‘𝑧) = (𝐹‘(𝐹‘𝑦)))
20		2fveq3 6874	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐹‘(𝐹‘𝑥)) = (𝐹‘(𝐹‘𝑦)))
21		id 22	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
22	20, 21	eqeq12d 2780	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝐹‘(𝐹‘𝑥)) = 𝑥 ↔ (𝐹‘(𝐹‘𝑦)) = 𝑦))
23		simplr 778	. . . . . . 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 3584	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = (𝐹‘𝑦))) → (𝐹‘(𝐹‘𝑦)) = 𝑦)
25	19, 24	eqtr2d 2800	. . . . 5 ⊢ (((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = (𝐹‘𝑦))) → 𝑦 = (𝐹‘𝑧))
26	18, 25	jca 519	. . . 4 ⊢ (((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = (𝐹‘𝑦))) → (𝑧 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑧)))
27	13, 26	impbida 810	. . 3 ⊢ ((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) → ((𝑧 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑧)) ↔ (𝑦 ∈ 𝐴 ∧ 𝑧 = (𝐹‘𝑦))))
28	27	mptcnv 6127	. 2 ⊢ ((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) → ◡(𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧)) = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)))
29		ffn 6693	. . . 4 ⊢ (𝐹:𝐴⟶𝐴 → 𝐹 Fn 𝐴)
30		dffn5 6927	. . . . 5 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧)))
31	30	birani 507	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) → 𝐹 = (𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧)))
32	29, 31	sylan 589	. . 3 ⊢ ((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) → 𝐹 = (𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧)))
33	32	cnveqd 5849	. 2 ⊢ ((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) → ◡𝐹 = ◡(𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧)))
34		dffn5 6927	. . . 4 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)))
35	34	birani 507	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) → 𝐹 = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)))
36	29, 35	sylan 589	. 2 ⊢ ((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) → 𝐹 = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)))
37	28, 33, 36	3eqtr4d 2809	1 ⊢ ((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) → ◡𝐹 = 𝐹)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1562   ∈ wcel 2144  ∀wral 3078   ↦ cmpt 5183  ^◡ccnv 5648   Fn wfn 6518  ⟶wf 6519  ‘cfv 6523

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-fv 6531

This theorem is referenced by:  mirf1o  28844  lmif1o  28970  nvocnvb  44003