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Theorem rinvf1o 32496
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 6755 . . . . 5 (Fun 𝐹𝐹:dom 𝐹⟶ran 𝐹)
31, 2mpbi 229 . . . 4 𝐹:dom 𝐹⟶ran 𝐹
4 rinvbij.2 . . . . . 6 𝐹 = 𝐹
54funeqi 6575 . . . . 5 (Fun 𝐹 ↔ Fun 𝐹)
61, 5mpbir 230 . . . 4 Fun 𝐹
7 df-f1 6554 . . . 4 (𝐹:dom 𝐹1-1→ran 𝐹 ↔ (𝐹:dom 𝐹⟶ran 𝐹 ∧ Fun 𝐹))
83, 6, 7mpbir2an 709 . . 3 𝐹:dom 𝐹1-1→ran 𝐹
9 rinvbij.4a . . 3 𝐴 ⊆ dom 𝐹
10 f1ores 6852 . . 3 ((𝐹:dom 𝐹1-1→ran 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴):𝐴1-1-onto→(𝐹𝐴))
118, 9, 10mp2an 690 . 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 7062 . . . . . . 7 ((Fun 𝐹𝐵 ⊆ dom 𝐹) → ((𝐹𝐵) ⊆ 𝐴𝐵 ⊆ (𝐹𝐴)))
161, 14, 15mp2an 690 . . . . . 6 ((𝐹𝐵) ⊆ 𝐴𝐵 ⊆ (𝐹𝐴))
1713, 16mpbi 229 . . . . 5 𝐵 ⊆ (𝐹𝐴)
184imaeq1i 6061 . . . . 5 (𝐹𝐴) = (𝐹𝐴)
1917, 18sseqtri 4013 . . . 4 𝐵 ⊆ (𝐹𝐴)
2012, 19eqssi 3993 . . 3 (𝐹𝐴) = 𝐵
21 f1oeq3 6828 . . 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 229 1 (𝐹𝐴):𝐴1-1-onto𝐵
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1533  wss 3944  ccnv 5677  dom cdm 5678  ran crn 5679  cres 5680  cima 5681  Fun wfun 6543  wf 6545  1-1wf1 6546  1-1-ontowf1o 6548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557
This theorem is referenced by:  ballotlem7  34286
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