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Theorem f1ocnv 5164
Description: The converse of a one-to-one onto function is also one-to-one onto. (Contributed by NM, 11-Feb-1997.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
Assertion
Ref Expression
f1ocnv (𝐹:𝐴1-1-onto𝐵𝐹:𝐵1-1-onto𝐴)

Proof of Theorem f1ocnv
StepHypRef Expression
1 fnrel 5022 . . . . 5 (𝐹 Fn 𝐴 → Rel 𝐹)
2 dfrel2 4796 . . . . . 6 (Rel 𝐹𝐹 = 𝐹)
3 fneq1 5012 . . . . . . 7 (𝐹 = 𝐹 → (𝐹 Fn 𝐴𝐹 Fn 𝐴))
43biimprd 151 . . . . . 6 (𝐹 = 𝐹 → (𝐹 Fn 𝐴𝐹 Fn 𝐴))
52, 4sylbi 118 . . . . 5 (Rel 𝐹 → (𝐹 Fn 𝐴𝐹 Fn 𝐴))
61, 5mpcom 36 . . . 4 (𝐹 Fn 𝐴𝐹 Fn 𝐴)
76anim2i 328 . . 3 ((𝐹 Fn 𝐵𝐹 Fn 𝐴) → (𝐹 Fn 𝐵𝐹 Fn 𝐴))
87ancoms 259 . 2 ((𝐹 Fn 𝐴𝐹 Fn 𝐵) → (𝐹 Fn 𝐵𝐹 Fn 𝐴))
9 dff1o4 5159 . 2 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹 Fn 𝐴𝐹 Fn 𝐵))
10 dff1o4 5159 . 2 (𝐹:𝐵1-1-onto𝐴 ↔ (𝐹 Fn 𝐵𝐹 Fn 𝐴))
118, 9, 103imtr4i 194 1 (𝐹:𝐴1-1-onto𝐵𝐹:𝐵1-1-onto𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1257  ccnv 4369  Rel wrel 4375   Fn wfn 4922  1-1-ontowf1o 4926
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-14 1419  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-sep 3900  ax-pow 3952  ax-pr 3969
This theorem depends on definitions:  df-bi 114  df-3an 896  df-tru 1260  df-nf 1364  df-sb 1660  df-eu 1917  df-mo 1918  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ral 2326  df-rex 2327  df-v 2574  df-un 2947  df-in 2949  df-ss 2956  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-br 3790  df-opab 3844  df-xp 4376  df-rel 4377  df-cnv 4378  df-co 4379  df-dm 4380  df-rn 4381  df-fun 4929  df-fn 4930  df-f 4931  df-f1 4932  df-fo 4933  df-f1o 4934
This theorem is referenced by:  f1ocnvb  5165  f1orescnv  5167  f1imacnv  5168  f1cnv  5175  f1ococnv1  5180  f1oresrab  5354  f1ocnvfv2  5443  f1ocnvdm  5446  f1ocnvfvrneq  5447  fcof1o  5454  isocnv  5476  f1ofveu  5525  ener  6287  en0  6303  en1  6307  ordiso2  6385  cnrecnv  9702
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