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Theorem f1ocnv 5052
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 4911 . . . . 5 (𝐹 Fn A → Rel 𝐹)
2 dfrel2 4686 . . . . . 6 (Rel 𝐹𝐹 = 𝐹)
3 fneq1 4901 . . . . . . 7 (𝐹 = 𝐹 → (𝐹 Fn A𝐹 Fn A))
43biimprd 147 . . . . . 6 (𝐹 = 𝐹 → (𝐹 Fn A𝐹 Fn A))
52, 4sylbi 114 . . . . 5 (Rel 𝐹 → (𝐹 Fn A𝐹 Fn A))
61, 5mpcom 32 . . . 4 (𝐹 Fn A𝐹 Fn A)
76anim2i 324 . . 3 ((𝐹 Fn B 𝐹 Fn A) → (𝐹 Fn B 𝐹 Fn A))
87ancoms 255 . 2 ((𝐹 Fn A 𝐹 Fn B) → (𝐹 Fn B 𝐹 Fn A))
9 dff1o4 5047 . 2 (𝐹:A1-1-ontoB ↔ (𝐹 Fn A 𝐹 Fn B))
10 dff1o4 5047 . 2 (𝐹:B1-1-ontoA ↔ (𝐹 Fn B 𝐹 Fn A))
118, 9, 103imtr4i 190 1 (𝐹:A1-1-ontoB𝐹:B1-1-ontoA)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1223  ccnv 4259  Rel wrel 4265   Fn wfn 4812  1-1-ontowf1o 4816
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 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-14 1378  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995  ax-sep 3838  ax-pow 3890  ax-pr 3907
This theorem depends on definitions:  df-bi 110  df-3an 869  df-tru 1226  df-nf 1323  df-sb 1619  df-eu 1876  df-mo 1877  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-ral 2280  df-rex 2281  df-v 2528  df-un 2890  df-in 2892  df-ss 2899  df-pw 3325  df-sn 3345  df-pr 3346  df-op 3348  df-br 3728  df-opab 3782  df-xp 4266  df-rel 4267  df-cnv 4268  df-co 4269  df-dm 4270  df-rn 4271  df-fun 4819  df-fn 4820  df-f 4821  df-f1 4822  df-fo 4823  df-f1o 4824
This theorem is referenced by:  f1ocnvb  5053  f1orescnv  5055  f1imacnv  5056  f1cnv  5063  f1ococnv1  5068  f1oresrab  5242  f1ocnvfv2  5331  f1ocnvdm  5334  f1ocnvfvrneq  5335  fcof1o  5342  isocnv  5364  f1ofveu  5412
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