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Theorem f1ocnvb 5083
Description: A relation is a one-to-one onto function iff its converse is a one-to-one onto function with domain and range interchanged. (Contributed by NM, 8-Dec-2003.)
Assertion
Ref Expression
f1ocnvb (Rel 𝐹 → (𝐹:A1-1-ontoB𝐹:B1-1-ontoA))

Proof of Theorem f1ocnvb
StepHypRef Expression
1 f1ocnv 5082 . 2 (𝐹:A1-1-ontoB𝐹:B1-1-ontoA)
2 f1ocnv 5082 . . 3 (𝐹:B1-1-ontoA𝐹:A1-1-ontoB)
3 dfrel2 4714 . . . 4 (Rel 𝐹𝐹 = 𝐹)
4 f1oeq1 5060 . . . 4 (𝐹 = 𝐹 → (𝐹:A1-1-ontoB𝐹:A1-1-ontoB))
53, 4sylbi 114 . . 3 (Rel 𝐹 → (𝐹:A1-1-ontoB𝐹:A1-1-ontoB))
62, 5syl5ib 143 . 2 (Rel 𝐹 → (𝐹:B1-1-ontoA𝐹:A1-1-ontoB))
71, 6impbid2 131 1 (Rel 𝐹 → (𝐹:A1-1-ontoB𝐹:B1-1-ontoA))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1242  ccnv 4287  Rel wrel 4293  1-1-ontowf1o 4844
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852
This theorem is referenced by: (None)
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