Theorem nvof1o 7278

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 6635	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
2		fdmrn 6737	. . . . . 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 6638	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
6	5	adantr 485	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) → dom 𝐹 = 𝐴)
7		df-rn 5672	. . . . . . 7 ⊢ ran 𝐹 = dom ◡𝐹
8		dmeq 5893	. . . . . . 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 6700	. . . 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 6556	. . . . 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 6541	. . 3 ⊢ (𝐹:𝐴–1-1→𝐴 ↔ (𝐹:𝐴⟶𝐴 ∧ Fun ◡𝐹))
18	12, 16, 17	sylanbrc 594	. 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 6542	. . 3 ⊢ (𝐹:𝐴–onto→𝐴 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐴))
21	19, 10, 20	sylanbrc 594	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) → 𝐹:𝐴–onto→𝐴)
22		df-f1o 6543	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐴 ↔ (𝐹:𝐴–1-1→𝐴 ∧ 𝐹:𝐴–onto→𝐴))
23	18, 21, 22	sylanbrc 594	1 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) → 𝐹:𝐴–1-1-onto→𝐴)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570  ^◡ccnv 5660  dom cdm 5661  ran crn 5662  Fun wfun 6530   Fn wfn 6531  ⟶wf 6532  –1-1→wf1 6533  –onto→wfo 6534  –1-1-onto→wf1o 6535

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543

This theorem is used by:  mirf1o  28955  lmif1o  29113  nvocnvb  44176  dssmapf1od  44775