Theorem nvof1o 7288

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 6639	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
2		fdmrn 6741	. . . . . 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 486	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) → 𝐹:dom 𝐹⟶ran 𝐹)
5		fndm 6642	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
6	5	adantr 486	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) → dom 𝐹 = 𝐴)
7		df-rn 5661	. . . . . . 7 ⊢ ran 𝐹 = dom ◡𝐹
8		dmeq 5884	. . . . . . 7 ⊢ (◡𝐹 = 𝐹 → dom ◡𝐹 = dom 𝐹)
9	7, 8	eqtrid 2807	. . . . . 6 ⊢ (◡𝐹 = 𝐹 → ran 𝐹 = dom 𝐹)
10	9, 5	sylan9eqr 2817	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) → ran 𝐹 = 𝐴)
11	6, 10	feq23d 6704	. . . 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 486	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) → Fun 𝐹)
14		funeq 6559	. . . . 5 ⊢ (◡𝐹 = 𝐹 → (Fun ◡𝐹 ↔ Fun 𝐹))
15	14	adantl 487	. . . 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 6543	. . 3 ⊢ (𝐹:𝐴–1-1→𝐴 ↔ (𝐹:𝐴⟶𝐴 ∧ Fun ◡𝐹))
18	12, 16, 17	sylanbrc 595	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) → 𝐹:𝐴–1-1→𝐴)
19		simpl 488	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) → 𝐹 Fn 𝐴)
20		df-fo 6544	. . 3 ⊢ (𝐹:𝐴–onto→𝐴 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐴))
21	19, 10, 20	sylanbrc 595	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) → 𝐹:𝐴–onto→𝐴)
22		df-f1o 6545	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐴 ↔ (𝐹:𝐴–1-1→𝐴 ∧ 𝐹:𝐴–onto→𝐴))
23	18, 21, 22	sylanbrc 595	1 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) → 𝐹:𝐴–1-1-onto→𝐴)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569  ^◡ccnv 5649  dom cdm 5650  ran crn 5651  Fun wfun 6532   Fn wfn 6533  ⟶wf 6534  –1-1→wf1 6535  –onto→wfo 6536  –1-1-onto→wf1o 6537

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545

This theorem is used by:  mirf1o  29141  lmif1o  29300  nvocnvb  44423  dssmapf1od  45021