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Theorem rinvf1o 24042
 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
rinvbij.2
rinvbij.3a
rinvbij.3b
rinvbij.4a
rinvbij.4b
Assertion
Ref Expression
rinvf1o

Proof of Theorem rinvf1o
StepHypRef Expression
1 rinvbij.1 . . . . 5
2 fdmrn 24039 . . . . 5
31, 2mpbi 200 . . . 4
4 rinvbij.2 . . . . . 6
54funeqi 5474 . . . . 5
61, 5mpbir 201 . . . 4
7 df-f1 5459 . . . 4
83, 6, 7mpbir2an 887 . . 3
9 rinvbij.4a . . 3
10 f1ores 5689 . . 3
118, 9, 10mp2an 654 . 2
12 rinvbij.3a . . . 4
13 rinvbij.3b . . . . . 6
14 rinvbij.4b . . . . . . 7
15 funimass3 5846 . . . . . . 7
161, 14, 15mp2an 654 . . . . . 6
1713, 16mpbi 200 . . . . 5
184imaeq1i 5200 . . . . 5
1917, 18sseqtri 3380 . . . 4
2012, 19eqssi 3364 . . 3
21 f1oeq3 5667 . . 3
2220, 21ax-mp 8 . 2
2311, 22mpbi 200 1
 Colors of variables: wff set class Syntax hints:   wb 177   wceq 1652   wss 3320  ccnv 4877   cdm 4878   crn 4879   cres 4880  cima 4881   wfun 5448  wf 5450  wf1 5451  wf1o 5453 This theorem is referenced by:  ballotlem7  24793 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462
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