Theorem nvof1o 7030

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 6435	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
2		fdmrn 6524	. . . . . 6 ⊢ (Fun 𝐹 ↔ 𝐹:dom 𝐹⟶ran 𝐹)
3	1, 2	sylib 221	. . . . 5 ⊢ (𝐹 Fn 𝐴 → 𝐹:dom 𝐹⟶ran 𝐹)
4	3	adantr 485	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) → 𝐹:dom 𝐹⟶ran 𝐹)
5		fndm 6437	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
6	5	adantr 485	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) → dom 𝐹 = 𝐴)
7		df-rn 5536	. . . . . . 7 ⊢ ran 𝐹 = dom ◡𝐹
8		dmeq 5744	. . . . . . 7 ⊢ (◡𝐹 = 𝐹 → dom ◡𝐹 = dom 𝐹)
9	7, 8	syl5eq 2806	. . . . . 6 ⊢ (◡𝐹 = 𝐹 → ran 𝐹 = dom 𝐹)
10	9, 5	sylan9eqr 2816	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) → ran 𝐹 = 𝐴)
11	6, 10	feq23d 6494	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) → (𝐹:dom 𝐹⟶ran 𝐹 ↔ 𝐹:𝐴⟶𝐴))
12	4, 11	mpbid 235	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) → 𝐹:𝐴⟶𝐴)
13	1	adantr 485	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) → Fun 𝐹)
14		funeq 6356	. . . . 5 ⊢ (◡𝐹 = 𝐹 → (Fun ◡𝐹 ↔ Fun 𝐹))
15	14	adantl 486	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) → (Fun ◡𝐹 ↔ Fun 𝐹))
16	13, 15	mpbird 260	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) → Fun ◡𝐹)
17		df-f1 6341	. . 3 ⊢ (𝐹:𝐴–1-1→𝐴 ↔ (𝐹:𝐴⟶𝐴 ∧ Fun ◡𝐹))
18	12, 16, 17	sylanbrc 587	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) → 𝐹:𝐴–1-1→𝐴)
19		simpl 487	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) → 𝐹 Fn 𝐴)
20		df-fo 6342	. . 3 ⊢ (𝐹:𝐴–onto→𝐴 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐴))
21	19, 10, 20	sylanbrc 587	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) → 𝐹:𝐴–onto→𝐴)
22		df-f1o 6343	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐴 ↔ (𝐹:𝐴–1-1→𝐴 ∧ 𝐹:𝐴–onto→𝐴))
23	18, 21, 22	sylanbrc 587	1 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) → 𝐹:𝐴–1-1-onto→𝐴)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1539  ^◡ccnv 5524  dom cdm 5525  ran crn 5526  Fun wfun 6330   Fn wfn 6331  ⟶wf 6332  –1-1→wf1 6333  –onto→wfo 6334  –1-1-onto→wf1o 6335

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-ex 1783  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-v 3412  df-un 3864  df-in 3866  df-ss 3876  df-sn 4524  df-pr 4526  df-op 4530  df-br 5034  df-opab 5096  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343

This theorem is referenced by:  mirf1o  26555  lmif1o  26681  dssmapf1od  41088