Theorem nvof1o 7226

Description: An involution is a bijection. (Contributed by Thierry Arnoux, 7-Dec-2016.)

Assertion

Ref	Expression
nvof1o	⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) → 𝐹:𝐴–1-1-onto→𝐴)

Proof of Theorem nvof1o

Step	Hyp	Ref	Expression
1		fnfun 6591	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
2		fdmrn 6692	. . . . . 6 ⊢ (Fun 𝐹 ↔ 𝐹:dom 𝐹⟶ran 𝐹)
3	1, 2	sylib 218	. . . . 5 ⊢ (𝐹 Fn 𝐴 → 𝐹:dom 𝐹⟶ran 𝐹)
4	3	adantr 480	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) → 𝐹:dom 𝐹⟶ran 𝐹)
5		fndm 6594	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
6	5	adantr 480	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) → dom 𝐹 = 𝐴)
7		df-rn 5634	. . . . . . 7 ⊢ ran 𝐹 = dom ◡𝐹
8		dmeq 5851	. . . . . . 7 ⊢ (◡𝐹 = 𝐹 → dom ◡𝐹 = dom 𝐹)
9	7, 8	eqtrid 2782	. . . . . 6 ⊢ (◡𝐹 = 𝐹 → ran 𝐹 = dom 𝐹)
10	9, 5	sylan9eqr 2792	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) → ran 𝐹 = 𝐴)
11	6, 10	feq23d 6656	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) → (𝐹:dom 𝐹⟶ran 𝐹 ↔ 𝐹:𝐴⟶𝐴))
12	4, 11	mpbid 232	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) → 𝐹:𝐴⟶𝐴)
13	1	adantr 480	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) → Fun 𝐹)
14		funeq 6511	. . . . 5 ⊢ (◡𝐹 = 𝐹 → (Fun ◡𝐹 ↔ Fun 𝐹))
15	14	adantl 481	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) → (Fun ◡𝐹 ↔ Fun 𝐹))
16	13, 15	mpbird 257	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) → Fun ◡𝐹)
17		df-f1 6496	. . 3 ⊢ (𝐹:𝐴–1-1→𝐴 ↔ (𝐹:𝐴⟶𝐴 ∧ Fun ◡𝐹))
18	12, 16, 17	sylanbrc 584	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) → 𝐹:𝐴–1-1→𝐴)
19		simpl 482	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) → 𝐹 Fn 𝐴)
20		df-fo 6497	. . 3 ⊢ (𝐹:𝐴–onto→𝐴 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐴))
21	19, 10, 20	sylanbrc 584	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) → 𝐹:𝐴–onto→𝐴)
22		df-f1o 6498	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐴 ↔ (𝐹:𝐴–1-1→𝐴 ∧ 𝐹:𝐴–onto→𝐴))
23	18, 21, 22	sylanbrc 584	1 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) → 𝐹:𝐴–1-1-onto→𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542  ^◡ccnv 5622  dom cdm 5623  ran crn 5624  Fun wfun 6485   Fn wfn 6486  ⟶wf 6487  –1-1→wf1 6488  –onto→wfo 6489  –1-1-onto→wf1o 6490

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-rab 3399  df-v 3441  df-dif 3903  df-un 3905  df-ss 3917  df-nul 4285  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5098  df-opab 5160  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498

This theorem is referenced by:  mirf1o  28722  lmif1o  28848  nvocnvb  43700  dssmapf1od  44299