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Mirrors > Home > MPE Home > Th. List > Mathboxes > rinvf1o | Structured version Visualization version GIF version |
Description: Sufficient conditions for the restriction of an involution to be a bijection. (Contributed by Thierry Arnoux, 7-Dec-2016.) |
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 |
Ref | Expression |
---|---|
rinvf1o | ⊢ (𝐹 ↾ 𝐴):𝐴–1-1-onto→𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rinvbij.1 | . . . . 5 ⊢ Fun 𝐹 | |
2 | fdmrn 6064 | . . . . 5 ⊢ (Fun 𝐹 ↔ 𝐹:dom 𝐹⟶ran 𝐹) | |
3 | 1, 2 | mpbi 220 | . . . 4 ⊢ 𝐹:dom 𝐹⟶ran 𝐹 |
4 | rinvbij.2 | . . . . . 6 ⊢ ◡𝐹 = 𝐹 | |
5 | 4 | funeqi 5909 | . . . . 5 ⊢ (Fun ◡𝐹 ↔ Fun 𝐹) |
6 | 1, 5 | mpbir 221 | . . . 4 ⊢ Fun ◡𝐹 |
7 | df-f1 5893 | . . . 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 6151 | . . 3 ⊢ ((𝐹:dom 𝐹–1-1→ran 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 ↾ 𝐴):𝐴–1-1-onto→(𝐹 “ 𝐴)) | |
11 | 8, 9, 10 | mp2an 708 | . 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 6333 | . . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐵 ⊆ dom 𝐹) → ((𝐹 “ 𝐵) ⊆ 𝐴 ↔ 𝐵 ⊆ (◡𝐹 “ 𝐴))) | |
16 | 1, 14, 15 | mp2an 708 | . . . . . 6 ⊢ ((𝐹 “ 𝐵) ⊆ 𝐴 ↔ 𝐵 ⊆ (◡𝐹 “ 𝐴)) |
17 | 13, 16 | mpbi 220 | . . . . 5 ⊢ 𝐵 ⊆ (◡𝐹 “ 𝐴) |
18 | 4 | imaeq1i 5463 | . . . . 5 ⊢ (◡𝐹 “ 𝐴) = (𝐹 “ 𝐴) |
19 | 17, 18 | sseqtri 3637 | . . . 4 ⊢ 𝐵 ⊆ (𝐹 “ 𝐴) |
20 | 12, 19 | eqssi 3619 | . . 3 ⊢ (𝐹 “ 𝐴) = 𝐵 |
21 | f1oeq3 6129 | . . 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 220 | 1 ⊢ (𝐹 ↾ 𝐴):𝐴–1-1-onto→𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 = wceq 1483 ⊆ wss 3574 ◡ccnv 5113 dom cdm 5114 ran crn 5115 ↾ cres 5116 “ cima 5117 Fun wfun 5882 ⟶wf 5884 –1-1→wf1 5885 –1-1-onto→wf1o 5887 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 |
This theorem is referenced by: ballotlem7 30597 |
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