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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 6367 | . . . . 5 ⊢ (Fun 𝐹 ↔ 𝐹:dom 𝐹⟶ran 𝐹) | |
3 | 1, 2 | mpbi 222 | . . . 4 ⊢ 𝐹:dom 𝐹⟶ran 𝐹 |
4 | rinvbij.2 | . . . . . 6 ⊢ ◡𝐹 = 𝐹 | |
5 | 4 | funeqi 6209 | . . . . 5 ⊢ (Fun ◡𝐹 ↔ Fun 𝐹) |
6 | 1, 5 | mpbir 223 | . . . 4 ⊢ Fun ◡𝐹 |
7 | df-f1 6193 | . . . 4 ⊢ (𝐹:dom 𝐹–1-1→ran 𝐹 ↔ (𝐹:dom 𝐹⟶ran 𝐹 ∧ Fun ◡𝐹)) | |
8 | 3, 6, 7 | mpbir2an 698 | . . 3 ⊢ 𝐹:dom 𝐹–1-1→ran 𝐹 |
9 | rinvbij.4a | . . 3 ⊢ 𝐴 ⊆ dom 𝐹 | |
10 | f1ores 6458 | . . 3 ⊢ ((𝐹:dom 𝐹–1-1→ran 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 ↾ 𝐴):𝐴–1-1-onto→(𝐹 “ 𝐴)) | |
11 | 8, 9, 10 | mp2an 679 | . 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 6649 | . . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐵 ⊆ dom 𝐹) → ((𝐹 “ 𝐵) ⊆ 𝐴 ↔ 𝐵 ⊆ (◡𝐹 “ 𝐴))) | |
16 | 1, 14, 15 | mp2an 679 | . . . . . 6 ⊢ ((𝐹 “ 𝐵) ⊆ 𝐴 ↔ 𝐵 ⊆ (◡𝐹 “ 𝐴)) |
17 | 13, 16 | mpbi 222 | . . . . 5 ⊢ 𝐵 ⊆ (◡𝐹 “ 𝐴) |
18 | 4 | imaeq1i 5767 | . . . . 5 ⊢ (◡𝐹 “ 𝐴) = (𝐹 “ 𝐴) |
19 | 17, 18 | sseqtri 3893 | . . . 4 ⊢ 𝐵 ⊆ (𝐹 “ 𝐴) |
20 | 12, 19 | eqssi 3874 | . . 3 ⊢ (𝐹 “ 𝐴) = 𝐵 |
21 | f1oeq3 6435 | . . 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 222 | 1 ⊢ (𝐹 ↾ 𝐴):𝐴–1-1-onto→𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 = wceq 1507 ⊆ wss 3829 ◡ccnv 5406 dom cdm 5407 ran crn 5408 ↾ cres 5409 “ cima 5410 Fun wfun 6182 ⟶wf 6184 –1-1→wf1 6185 –1-1-onto→wf1o 6187 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5060 ax-nul 5067 ax-pr 5186 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-ral 3093 df-rex 3094 df-rab 3097 df-v 3417 df-sbc 3682 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-nul 4179 df-if 4351 df-sn 4442 df-pr 4444 df-op 4448 df-uni 4713 df-br 4930 df-opab 4992 df-id 5312 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 |
This theorem is referenced by: ballotlem7 31445 |
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