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Theorem f1cnvcnv 5152
Description: Two ways to express that a set 𝐴 (not necessarily a function) is one-to-one. Each side is equivalent to Definition 6.4(3) of [TakeutiZaring] p. 24, who use the notation "Un2 (A)" for one-to-one. We do not introduce a separate notation since we rarely use it. (Contributed by NM, 13-Aug-2004.)
Assertion
Ref Expression
f1cnvcnv (𝐴:dom 𝐴1-1→V ↔ (Fun 𝐴 ∧ Fun 𝐴))

Proof of Theorem f1cnvcnv
StepHypRef Expression
1 df-f1 4957 . 2 (𝐴:dom 𝐴1-1→V ↔ (𝐴:dom 𝐴⟶V ∧ Fun 𝐴))
2 dffn2 5098 . . . 4 (𝐴 Fn dom 𝐴𝐴:dom 𝐴⟶V)
3 dmcnvcnv 4606 . . . . 5 dom 𝐴 = dom 𝐴
4 df-fn 4955 . . . . 5 (𝐴 Fn dom 𝐴 ↔ (Fun 𝐴 ∧ dom 𝐴 = dom 𝐴))
53, 4mpbiran2 883 . . . 4 (𝐴 Fn dom 𝐴 ↔ Fun 𝐴)
62, 5bitr3i 184 . . 3 (𝐴:dom 𝐴⟶V ↔ Fun 𝐴)
7 relcnv 4753 . . . . 5 Rel 𝐴
8 dfrel2 4821 . . . . 5 (Rel 𝐴𝐴 = 𝐴)
97, 8mpbi 143 . . . 4 𝐴 = 𝐴
109funeqi 4972 . . 3 (Fun 𝐴 ↔ Fun 𝐴)
116, 10anbi12ci 449 . 2 ((𝐴:dom 𝐴⟶V ∧ Fun 𝐴) ↔ (Fun 𝐴 ∧ Fun 𝐴))
121, 11bitri 182 1 (𝐴:dom 𝐴1-1→V ↔ (Fun 𝐴 ∧ Fun 𝐴))
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103   = wceq 1285  Vcvv 2610  ccnv 4390  dom cdm 4391  Rel wrel 4396  Fun wfun 4946   Fn wfn 4947  wf 4948  1-1wf1 4949
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957
This theorem is referenced by: (None)
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