Theorem rinvf1o 32640

Description: Sufficient conditions for the restriction of an involution to be a bijection. (Contributed by Thierry Arnoux, 7-Dec-2016.)

Hypotheses

Ref	Expression
rinvbij.1	⊢ Fun 𝐹
rinvbij.2	⊢ ◡𝐹 = 𝐹
rinvbij.3a	⊢ (𝐹 “ 𝐴) ⊆ 𝐵
rinvbij.3b	⊢ (𝐹 “ 𝐵) ⊆ 𝐴
rinvbij.4a	⊢ 𝐴 ⊆ dom 𝐹
rinvbij.4b	⊢ 𝐵 ⊆ dom 𝐹

Assertion

Ref	Expression
rinvf1o	⊢ (𝐹 ↾ 𝐴):𝐴–1-1-onto→𝐵

Proof of Theorem rinvf1o

Step	Hyp	Ref	Expression
1		rinvbij.1	. . . . 5 ⊢ Fun 𝐹
2		fdmrn 6778	. . . . 5 ⊢ (Fun 𝐹 ↔ 𝐹:dom 𝐹⟶ran 𝐹)
3	1, 2	mpbi 230	. . . 4 ⊢ 𝐹:dom 𝐹⟶ran 𝐹
4		rinvbij.2	. . . . . 6 ⊢ ◡𝐹 = 𝐹
5	4	funeqi 6598	. . . . 5 ⊢ (Fun ◡𝐹 ↔ Fun 𝐹)
6	1, 5	mpbir 231	. . . 4 ⊢ Fun ◡𝐹
7		df-f1 6577	. . . 4 ⊢ (𝐹:dom 𝐹–1-1→ran 𝐹 ↔ (𝐹:dom 𝐹⟶ran 𝐹 ∧ Fun ◡𝐹))
8	3, 6, 7	mpbir2an 710	. . 3 ⊢ 𝐹:dom 𝐹–1-1→ran 𝐹
9		rinvbij.4a	. . 3 ⊢ 𝐴 ⊆ dom 𝐹
10		f1ores 6875	. . 3 ⊢ ((𝐹:dom 𝐹–1-1→ran 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 ↾ 𝐴):𝐴–1-1-onto→(𝐹 “ 𝐴))
11	8, 9, 10	mp2an 691	. 2 ⊢ (𝐹 ↾ 𝐴):𝐴–1-1-onto→(𝐹 “ 𝐴)
12		rinvbij.3a	. . . 4 ⊢ (𝐹 “ 𝐴) ⊆ 𝐵
13		rinvbij.3b	. . . . . 6 ⊢ (𝐹 “ 𝐵) ⊆ 𝐴
14		rinvbij.4b	. . . . . . 7 ⊢ 𝐵 ⊆ dom 𝐹
15		funimass3 7085	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐵 ⊆ dom 𝐹) → ((𝐹 “ 𝐵) ⊆ 𝐴 ↔ 𝐵 ⊆ (◡𝐹 “ 𝐴)))
16	1, 14, 15	mp2an 691	. . . . . 6 ⊢ ((𝐹 “ 𝐵) ⊆ 𝐴 ↔ 𝐵 ⊆ (◡𝐹 “ 𝐴))
17	13, 16	mpbi 230	. . . . 5 ⊢ 𝐵 ⊆ (◡𝐹 “ 𝐴)
18	4	imaeq1i 6085	. . . . 5 ⊢ (◡𝐹 “ 𝐴) = (𝐹 “ 𝐴)
19	17, 18	sseqtri 4039	. . . 4 ⊢ 𝐵 ⊆ (𝐹 “ 𝐴)
20	12, 19	eqssi 4019	. . 3 ⊢ (𝐹 “ 𝐴) = 𝐵
21		f1oeq3 6851	. . 3 ⊢ ((𝐹 “ 𝐴) = 𝐵 → ((𝐹 ↾ 𝐴):𝐴–1-1-onto→(𝐹 “ 𝐴) ↔ (𝐹 ↾ 𝐴):𝐴–1-1-onto→𝐵))
22	20, 21	ax-mp 5	. 2 ⊢ ((𝐹 ↾ 𝐴):𝐴–1-1-onto→(𝐹 “ 𝐴) ↔ (𝐹 ↾ 𝐴):𝐴–1-1-onto→𝐵)
23	11, 22	mpbi 230	1 ⊢ (𝐹 ↾ 𝐴):𝐴–1-1-onto→𝐵

Syntax hints:   ↔ wb 206   = wceq 1537   ⊆ wss 3970  ^◡ccnv 5698  dom cdm 5699  ran crn 5700   ↾ cres 5701   “ cima 5702  Fun wfun 6566  ⟶wf 6568  –1-1→wf1 6569  –1-1-onto→wf1o 6571

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pr 5450

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3439  df-v 3484  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5170  df-opab 5232  df-id 5597  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-iota 6524  df-fun 6574  df-fn 6575  df-f 6576  df-f1 6577  df-fo 6578  df-f1o 6579  df-fv 6580

This theorem is referenced by:  ballotlem7  34492