Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rinvf1o Structured version   Visualization version   GIF version

Theorem rinvf1o 29273
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 (𝐹𝐴) ⊆ 𝐵
rinvbij.3b (𝐹𝐵) ⊆ 𝐴
rinvbij.4a 𝐴 ⊆ dom 𝐹
rinvbij.4b 𝐵 ⊆ dom 𝐹
Assertion
Ref Expression
rinvf1o (𝐹𝐴):𝐴1-1-onto𝐵

Proof of Theorem rinvf1o
StepHypRef Expression
1 rinvbij.1 . . . . 5 Fun 𝐹
2 fdmrn 6021 . . . . 5 (Fun 𝐹𝐹:dom 𝐹⟶ran 𝐹)
31, 2mpbi 220 . . . 4 𝐹:dom 𝐹⟶ran 𝐹
4 rinvbij.2 . . . . . 6 𝐹 = 𝐹
54funeqi 5868 . . . . 5 (Fun 𝐹 ↔ Fun 𝐹)
61, 5mpbir 221 . . . 4 Fun 𝐹
7 df-f1 5852 . . . 4 (𝐹:dom 𝐹1-1→ran 𝐹 ↔ (𝐹:dom 𝐹⟶ran 𝐹 ∧ Fun 𝐹))
83, 6, 7mpbir2an 954 . . 3 𝐹:dom 𝐹1-1→ran 𝐹
9 rinvbij.4a . . 3 𝐴 ⊆ dom 𝐹
10 f1ores 6108 . . 3 ((𝐹:dom 𝐹1-1→ran 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴):𝐴1-1-onto→(𝐹𝐴))
118, 9, 10mp2an 707 . 2 (𝐹𝐴):𝐴1-1-onto→(𝐹𝐴)
12 rinvbij.3a . . . 4 (𝐹𝐴) ⊆ 𝐵
13 rinvbij.3b . . . . . 6 (𝐹𝐵) ⊆ 𝐴
14 rinvbij.4b . . . . . . 7 𝐵 ⊆ dom 𝐹
15 funimass3 6289 . . . . . . 7 ((Fun 𝐹𝐵 ⊆ dom 𝐹) → ((𝐹𝐵) ⊆ 𝐴𝐵 ⊆ (𝐹𝐴)))
161, 14, 15mp2an 707 . . . . . 6 ((𝐹𝐵) ⊆ 𝐴𝐵 ⊆ (𝐹𝐴))
1713, 16mpbi 220 . . . . 5 𝐵 ⊆ (𝐹𝐴)
184imaeq1i 5422 . . . . 5 (𝐹𝐴) = (𝐹𝐴)
1917, 18sseqtri 3616 . . . 4 𝐵 ⊆ (𝐹𝐴)
2012, 19eqssi 3599 . . 3 (𝐹𝐴) = 𝐵
21 f1oeq3 6086 . . 3 ((𝐹𝐴) = 𝐵 → ((𝐹𝐴):𝐴1-1-onto→(𝐹𝐴) ↔ (𝐹𝐴):𝐴1-1-onto𝐵))
2220, 21ax-mp 5 . 2 ((𝐹𝐴):𝐴1-1-onto→(𝐹𝐴) ↔ (𝐹𝐴):𝐴1-1-onto𝐵)
2311, 22mpbi 220 1 (𝐹𝐴):𝐴1-1-onto𝐵
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1480  wss 3555  ccnv 5073  dom cdm 5074  ran crn 5075  cres 5076  cima 5077  Fun wfun 5841  wf 5843  1-1wf1 5844  1-1-ontowf1o 5846
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855
This theorem is referenced by:  ballotlem7  30375
  Copyright terms: Public domain W3C validator