Theorem rinvf1o 33048

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 6741	. . . . 5 ⊢ (Fun 𝐹 ↔ 𝐹:dom 𝐹⟶ran 𝐹)
3	1, 2	mpbi 233	. . . 4 ⊢ 𝐹:dom 𝐹⟶ran 𝐹
4		rinvbij.2	. . . . . 6 ⊢ ◡𝐹 = 𝐹
5	4	funeqi 6561	. . . . 5 ⊢ (Fun ◡𝐹 ↔ Fun 𝐹)
6	1, 5	mpbir 234	. . . 4 ⊢ Fun ◡𝐹
7		df-f1 6545	. . . 4 ⊢ (𝐹:dom 𝐹–1-1→ran 𝐹 ↔ (𝐹:dom 𝐹⟶ran 𝐹 ∧ Fun ◡𝐹))
8	3, 6, 7	mpbir2an 724	. . 3 ⊢ 𝐹:dom 𝐹–1-1→ran 𝐹
9		rinvbij.4a	. . 3 ⊢ 𝐴 ⊆ dom 𝐹
10		f1ores 6839	. . 3 ⊢ ((𝐹:dom 𝐹–1-1→ran 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 ↾ 𝐴):𝐴–1-1-onto→(𝐹 “ 𝐴))
11	8, 9, 10	mp2an 705	. 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 7053	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐵 ⊆ dom 𝐹) → ((𝐹 “ 𝐵) ⊆ 𝐴 ↔ 𝐵 ⊆ (◡𝐹 “ 𝐴)))
16	1, 14, 15	mp2an 705	. . . . . 6 ⊢ ((𝐹 “ 𝐵) ⊆ 𝐴 ↔ 𝐵 ⊆ (◡𝐹 “ 𝐴))
17	13, 16	mpbi 233	. . . . 5 ⊢ 𝐵 ⊆ (◡𝐹 “ 𝐴)
18	4	imaeq1i 6061	. . . . 5 ⊢ (◡𝐹 “ 𝐴) = (𝐹 “ 𝐴)
19	17, 18	sseqtri 3986	. . . 4 ⊢ 𝐵 ⊆ (𝐹 “ 𝐴)
20	12, 19	eqssi 3954	. . 3 ⊢ (𝐹 “ 𝐴) = 𝐵
21		f1oeq3 6814	. . 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 233	1 ⊢ (𝐹 ↾ 𝐴):𝐴–1-1-onto→𝐵

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570   ⊆ wss 3906  ^◡ccnv 5662  dom cdm 5663  ran crn 5664   ↾ cres 5665   “ cima 5666  Fun wfun 6534  ⟶wf 6536  –1-1→wf1 6537  –1-1-onto→wf1o 6539

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

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-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548

This theorem is used by:  ballotlem7  34993