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Theorem rinvf1o 29763
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 6226 . . . . 5 (Fun 𝐹𝐹:dom 𝐹⟶ran 𝐹)
31, 2mpbi 220 . . . 4 𝐹:dom 𝐹⟶ran 𝐹
4 rinvbij.2 . . . . . 6 𝐹 = 𝐹
54funeqi 6071 . . . . 5 (Fun 𝐹 ↔ Fun 𝐹)
61, 5mpbir 221 . . . 4 Fun 𝐹
7 df-f1 6055 . . . 4 (𝐹:dom 𝐹1-1→ran 𝐹 ↔ (𝐹:dom 𝐹⟶ran 𝐹 ∧ Fun 𝐹))
83, 6, 7mpbir2an 993 . . 3 𝐹:dom 𝐹1-1→ran 𝐹
9 rinvbij.4a . . 3 𝐴 ⊆ dom 𝐹
10 f1ores 6314 . . 3 ((𝐹:dom 𝐹1-1→ran 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴):𝐴1-1-onto→(𝐹𝐴))
118, 9, 10mp2an 710 . 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 6498 . . . . . . 7 ((Fun 𝐹𝐵 ⊆ dom 𝐹) → ((𝐹𝐵) ⊆ 𝐴𝐵 ⊆ (𝐹𝐴)))
161, 14, 15mp2an 710 . . . . . 6 ((𝐹𝐵) ⊆ 𝐴𝐵 ⊆ (𝐹𝐴))
1713, 16mpbi 220 . . . . 5 𝐵 ⊆ (𝐹𝐴)
184imaeq1i 5622 . . . . 5 (𝐹𝐴) = (𝐹𝐴)
1917, 18sseqtri 3779 . . . 4 𝐵 ⊆ (𝐹𝐴)
2012, 19eqssi 3761 . . 3 (𝐹𝐴) = 𝐵
21 f1oeq3 6292 . . 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 1632  wss 3716  ccnv 5266  dom cdm 5267  ran crn 5268  cres 5269  cima 5270  Fun wfun 6044  wf 6046  1-1wf1 6047  1-1-ontowf1o 6049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pr 5056
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-sbc 3578  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-opab 4866  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058
This theorem is referenced by:  ballotlem7  30928
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