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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 6541 | . . . . 5 ⊢ (Fun 𝐹 ↔ 𝐹:dom 𝐹⟶ran 𝐹) | |
3 | 1, 2 | mpbi 232 | . . . 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 233 | . . . 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 709 | . . 3 ⊢ 𝐹:dom 𝐹–1-1→ran 𝐹 |
9 | rinvbij.4a | . . 3 ⊢ 𝐴 ⊆ dom 𝐹 | |
10 | f1ores 6632 | . . 3 ⊢ ((𝐹:dom 𝐹–1-1→ran 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 ↾ 𝐴):𝐴–1-1-onto→(𝐹 “ 𝐴)) | |
11 | 8, 9, 10 | mp2an 690 | . 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 6827 | . . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐵 ⊆ dom 𝐹) → ((𝐹 “ 𝐵) ⊆ 𝐴 ↔ 𝐵 ⊆ (◡𝐹 “ 𝐴))) | |
16 | 1, 14, 15 | mp2an 690 | . . . . . 6 ⊢ ((𝐹 “ 𝐵) ⊆ 𝐴 ↔ 𝐵 ⊆ (◡𝐹 “ 𝐴)) |
17 | 13, 16 | mpbi 232 | . . . . 5 ⊢ 𝐵 ⊆ (◡𝐹 “ 𝐴) |
18 | 4 | imaeq1i 5929 | . . . . 5 ⊢ (◡𝐹 “ 𝐴) = (𝐹 “ 𝐴) |
19 | 17, 18 | sseqtri 4006 | . . . 4 ⊢ 𝐵 ⊆ (𝐹 “ 𝐴) |
20 | 12, 19 | eqssi 3986 | . . 3 ⊢ (𝐹 “ 𝐴) = 𝐵 |
21 | f1oeq3 6609 | . . 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 232 | 1 ⊢ (𝐹 ↾ 𝐴):𝐴–1-1-onto→𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1536 ⊆ wss 3939 ◡ccnv 5557 dom cdm 5558 ran crn 5559 ↾ cres 5560 “ cima 5561 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 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 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 31797 |
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