Theorem nvof1o 7308

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

Assertion

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

Proof of Theorem nvof1o

Step	Hyp	Ref	Expression
1		fnfun 6658	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
2		fdmrn 6760	. . . . . 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 6661	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
6	5	adantr 486	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) → dom 𝐹 = 𝐴)
7		df-rn 5666	. . . . . . 7 ⊢ ran 𝐹 = dom ◡𝐹
8		dmeq 5891	. . . . . . 7 ⊢ (◡𝐹 = 𝐹 → dom ◡𝐹 = dom 𝐹)
9	7, 8	eqtrid 2810	. . . . . 6 ⊢ (◡𝐹 = 𝐹 → ran 𝐹 = dom 𝐹)
10	9, 5	sylan9eqr 2820	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) → ran 𝐹 = 𝐴)
11	6, 10	feq23d 6723	. . . 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 6576	. . . . 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 6560	. . 3 ⊢ (𝐹:𝐴⟶1-1𝐴 ↔ (𝐹:𝐴⟶𝐴 ∧ Fun ◡𝐹))
18	12, 16, 17	sylanbrc 595	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) → 𝐹:𝐴⟶1-1𝐴)
19		simpl 488	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) → 𝐹 Fn 𝐴)
20		df-fo 6561	. . 3 ⊢ (𝐹:𝐴⟶onto𝐴 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐴))
21	19, 10, 20	sylanbrc 595	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) → 𝐹:𝐴⟶onto𝐴)
22		df-f1o 6562	. 2 ⊢ (𝐹:𝐴⟶1-1onto𝐴 ↔ (𝐹:𝐴⟶1-1𝐴 ∧ 𝐹:𝐴⟶onto𝐴))
23	18, 21, 22	sylanbrc 595	1 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) → 𝐹:𝐴⟶1-1onto𝐴)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ^◡ccnv 5654  dom cdm 5655  ran crn 5656  Fun wfun 6549   Fn wfn 6550  ⟶wf 6551  ⟶1-1wf1 6552  ⟶ontowfo 6553  ⟶1-1ontowf1o 6554

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562

This theorem is used by:  mirf1o  29168  lmif1o  29327  nvocnvb  44613  dssmapf1od  45211