Theorem rinvf1o 30965

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 6632	. . . . 5 ⊢ (Fun 𝐹 ↔ 𝐹:dom 𝐹⟶ran 𝐹)
3	1, 2	mpbi 229	. . . 4 ⊢ 𝐹:dom 𝐹⟶ran 𝐹
4		rinvbij.2	. . . . . 6 ⊢ ◡𝐹 = 𝐹
5	4	funeqi 6455	. . . . 5 ⊢ (Fun ◡𝐹 ↔ Fun 𝐹)
6	1, 5	mpbir 230	. . . 4 ⊢ Fun ◡𝐹
7		df-f1 6438	. . . 4 ⊢ (𝐹:dom 𝐹–1-1→ran 𝐹 ↔ (𝐹:dom 𝐹⟶ran 𝐹 ∧ Fun ◡𝐹))
8	3, 6, 7	mpbir2an 708	. . 3 ⊢ 𝐹:dom 𝐹–1-1→ran 𝐹
9		rinvbij.4a	. . 3 ⊢ 𝐴 ⊆ dom 𝐹
10		f1ores 6730	. . 3 ⊢ ((𝐹:dom 𝐹–1-1→ran 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 ↾ 𝐴):𝐴–1-1-onto→(𝐹 “ 𝐴))
11	8, 9, 10	mp2an 689	. 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 6931	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐵 ⊆ dom 𝐹) → ((𝐹 “ 𝐵) ⊆ 𝐴 ↔ 𝐵 ⊆ (◡𝐹 “ 𝐴)))
16	1, 14, 15	mp2an 689	. . . . . 6 ⊢ ((𝐹 “ 𝐵) ⊆ 𝐴 ↔ 𝐵 ⊆ (◡𝐹 “ 𝐴))
17	13, 16	mpbi 229	. . . . 5 ⊢ 𝐵 ⊆ (◡𝐹 “ 𝐴)
18	4	imaeq1i 5966	. . . . 5 ⊢ (◡𝐹 “ 𝐴) = (𝐹 “ 𝐴)
19	17, 18	sseqtri 3957	. . . 4 ⊢ 𝐵 ⊆ (𝐹 “ 𝐴)
20	12, 19	eqssi 3937	. . 3 ⊢ (𝐹 “ 𝐴) = 𝐵
21		f1oeq3 6706	. . 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 229	1 ⊢ (𝐹 ↾ 𝐴):𝐴–1-1-onto→𝐵

Syntax hints:   ↔ wb 205   = wceq 1539   ⊆ wss 3887  ^◡ccnv 5588  dom cdm 5589  ran crn 5590   ↾ cres 5591   “ cima 5592  Fun wfun 6427  ⟶wf 6429  –1-1→wf1 6430  –1-1-onto→wf1o 6432

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441

This theorem is referenced by:  ballotlem7  32502