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Theorem rinvf1o 30138
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 6367 . . . . 5 (Fun 𝐹𝐹:dom 𝐹⟶ran 𝐹)
31, 2mpbi 222 . . . 4 𝐹:dom 𝐹⟶ran 𝐹
4 rinvbij.2 . . . . . 6 𝐹 = 𝐹
54funeqi 6209 . . . . 5 (Fun 𝐹 ↔ Fun 𝐹)
61, 5mpbir 223 . . . 4 Fun 𝐹
7 df-f1 6193 . . . 4 (𝐹:dom 𝐹1-1→ran 𝐹 ↔ (𝐹:dom 𝐹⟶ran 𝐹 ∧ Fun 𝐹))
83, 6, 7mpbir2an 698 . . 3 𝐹:dom 𝐹1-1→ran 𝐹
9 rinvbij.4a . . 3 𝐴 ⊆ dom 𝐹
10 f1ores 6458 . . 3 ((𝐹:dom 𝐹1-1→ran 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴):𝐴1-1-onto→(𝐹𝐴))
118, 9, 10mp2an 679 . 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 6649 . . . . . . 7 ((Fun 𝐹𝐵 ⊆ dom 𝐹) → ((𝐹𝐵) ⊆ 𝐴𝐵 ⊆ (𝐹𝐴)))
161, 14, 15mp2an 679 . . . . . 6 ((𝐹𝐵) ⊆ 𝐴𝐵 ⊆ (𝐹𝐴))
1713, 16mpbi 222 . . . . 5 𝐵 ⊆ (𝐹𝐴)
184imaeq1i 5767 . . . . 5 (𝐹𝐴) = (𝐹𝐴)
1917, 18sseqtri 3893 . . . 4 𝐵 ⊆ (𝐹𝐴)
2012, 19eqssi 3874 . . 3 (𝐹𝐴) = 𝐵
21 f1oeq3 6435 . . 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 222 1 (𝐹𝐴):𝐴1-1-onto𝐵
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  1-1wf1 6185  1-1-ontowf1o 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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