Theorem rinvf1o 6029

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 6028	. . . . 5 ⊢ (Fun 𝐹 ↔ 𝐹:dom 𝐹⟶ran 𝐹)
3	1, 2	mpbi 145	. . . 4 ⊢ 𝐹:dom 𝐹⟶ran 𝐹
4		rinvbij.2	. . . . . 6 ⊢ ◡𝐹 = 𝐹
5	4	funeqi 5396	. . . . 5 ⊢ (Fun ◡𝐹 ↔ Fun 𝐹)
6	1, 5	mpbir 146	. . . 4 ⊢ Fun ◡𝐹
7		df-f1 5380	. . . 4 ⊢ (𝐹:dom 𝐹–1-1→ran 𝐹 ↔ (𝐹:dom 𝐹⟶ran 𝐹 ∧ Fun ◡𝐹))
8	3, 6, 7	mpbir2an 955	. . 3 ⊢ 𝐹:dom 𝐹–1-1→ran 𝐹
9		rinvbij.4a	. . 3 ⊢ 𝐴 ⊆ dom 𝐹
10		f1ores 5652	. . 3 ⊢ ((𝐹:dom 𝐹–1-1→ran 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 ↾ 𝐴):𝐴–1-1-onto→(𝐹 “ 𝐴))
11	8, 9, 10	mp2an 430	. 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 5819	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐵 ⊆ dom 𝐹) → ((𝐹 “ 𝐵) ⊆ 𝐴 ↔ 𝐵 ⊆ (◡𝐹 “ 𝐴)))
16	1, 14, 15	mp2an 430	. . . . . 6 ⊢ ((𝐹 “ 𝐵) ⊆ 𝐴 ↔ 𝐵 ⊆ (◡𝐹 “ 𝐴))
17	13, 16	mpbi 145	. . . . 5 ⊢ 𝐵 ⊆ (◡𝐹 “ 𝐴)
18	4	imaeq1i 5121	. . . . 5 ⊢ (◡𝐹 “ 𝐴) = (𝐹 “ 𝐴)
19	17, 18	sseqtri 3282	. . . 4 ⊢ 𝐵 ⊆ (𝐹 “ 𝐴)
20	12, 19	eqssi 3264	. . 3 ⊢ (𝐹 “ 𝐴) = 𝐵
21		f1oeq3 5627	. . 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 145	1 ⊢ (𝐹 ↾ 𝐴):𝐴–1-1-onto→𝐵

Syntax hints:   ↔ wb 105   = wceq 1402   ⊆ wss 3220  ^◡ccnv 4771  dom cdm 4772  ran crn 4773   ↾ cres 4774   “ cima 4775  Fun wfun 5369  ⟶wf 5371  –1-1→wf1 5372  –1-1-onto→wf1o 5374

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383

This theorem is referenced by:  ballotfilem7  13262