Theorem rinvf1o 32554

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 6719	. . . . 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 6537	. . . . 5 ⊢ (Fun ◡𝐹 ↔ Fun 𝐹)
6	1, 5	mpbir 231	. . . 4 ⊢ Fun ◡𝐹
7		df-f1 6516	. . . 4 ⊢ (𝐹:dom 𝐹–1-1→ran 𝐹 ↔ (𝐹:dom 𝐹⟶ran 𝐹 ∧ Fun ◡𝐹))
8	3, 6, 7	mpbir2an 711	. . 3 ⊢ 𝐹:dom 𝐹–1-1→ran 𝐹
9		rinvbij.4a	. . 3 ⊢ 𝐴 ⊆ dom 𝐹
10		f1ores 6814	. . 3 ⊢ ((𝐹:dom 𝐹–1-1→ran 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 ↾ 𝐴):𝐴–1-1-onto→(𝐹 “ 𝐴))
11	8, 9, 10	mp2an 692	. 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 7026	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐵 ⊆ dom 𝐹) → ((𝐹 “ 𝐵) ⊆ 𝐴 ↔ 𝐵 ⊆ (◡𝐹 “ 𝐴)))
16	1, 14, 15	mp2an 692	. . . . . 6 ⊢ ((𝐹 “ 𝐵) ⊆ 𝐴 ↔ 𝐵 ⊆ (◡𝐹 “ 𝐴))
17	13, 16	mpbi 230	. . . . 5 ⊢ 𝐵 ⊆ (◡𝐹 “ 𝐴)
18	4	imaeq1i 6028	. . . . 5 ⊢ (◡𝐹 “ 𝐴) = (𝐹 “ 𝐴)
19	17, 18	sseqtri 3995	. . . 4 ⊢ 𝐵 ⊆ (𝐹 “ 𝐴)
20	12, 19	eqssi 3963	. . 3 ⊢ (𝐹 “ 𝐴) = 𝐵
21		f1oeq3 6790	. . 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 1540   ⊆ wss 3914  ^◡ccnv 5637  dom cdm 5638  ran crn 5639   ↾ cres 5640   “ cima 5641  Fun wfun 6505  ⟶wf 6507  –1-1→wf1 6508  –1-1-onto→wf1o 6510

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519

This theorem is referenced by:  ballotlem7  34527