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Theorem f1ocnvb 6454
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 6453 . 2 (𝐹:𝐴1-1-onto𝐵𝐹:𝐵1-1-onto𝐴)
2 f1ocnv 6453 . . 3 (𝐹:𝐵1-1-onto𝐴𝐹:𝐴1-1-onto𝐵)
3 dfrel2 5883 . . . 4 (Rel 𝐹𝐹 = 𝐹)
4 f1oeq1 6430 . . . 4 (𝐹 = 𝐹 → (𝐹:𝐴1-1-onto𝐵𝐹:𝐴1-1-onto𝐵))
53, 4sylbi 209 . . 3 (Rel 𝐹 → (𝐹:𝐴1-1-onto𝐵𝐹:𝐴1-1-onto𝐵))
62, 5syl5ib 236 . 2 (Rel 𝐹 → (𝐹:𝐵1-1-onto𝐴𝐹:𝐴1-1-onto𝐵))
71, 6impbid2 218 1 (Rel 𝐹 → (𝐹:𝐴1-1-onto𝐵𝐹:𝐵1-1-onto𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198   = wceq 1507  ccnv 5402  Rel wrel 5408  1-1-ontowf1o 6184
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 2744  ax-sep 5056  ax-nul 5063  ax-pr 5182
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 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-rab 3091  df-v 3411  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-br 4926  df-opab 4988  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192
This theorem is referenced by:  hasheqf1oi  13525
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