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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 6555 | . . . . 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 6379 | . . . . 5 ⊢ (Fun ◡𝐹 ↔ Fun 𝐹) |
6 | 1, 5 | mpbir 234 | . . . 4 ⊢ Fun ◡𝐹 |
7 | df-f1 6363 | . . . 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 6653 | . . 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 6852 | . . . . . . 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 233 | . . . . 5 ⊢ 𝐵 ⊆ (◡𝐹 “ 𝐴) |
18 | 4 | imaeq1i 5911 | . . . . 5 ⊢ (◡𝐹 “ 𝐴) = (𝐹 “ 𝐴) |
19 | 17, 18 | sseqtri 3923 | . . . 4 ⊢ 𝐵 ⊆ (𝐹 “ 𝐴) |
20 | 12, 19 | eqssi 3903 | . . 3 ⊢ (𝐹 “ 𝐴) = 𝐵 |
21 | f1oeq3 6629 | . . 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 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1543 ⊆ wss 3853 ◡ccnv 5535 dom cdm 5536 ran crn 5537 ↾ cres 5538 “ cima 5539 Fun wfun 6352 ⟶wf 6354 –1-1→wf1 6355 –1-1-onto→wf1o 6357 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 |
This theorem is referenced by: ballotlem7 32168 |
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