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Theorem f1cnvcnv 6814
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 6568 . 2 (𝐴:dom 𝐴1-1→V ↔ (𝐴:dom 𝐴⟶V ∧ Fun 𝐴))
2 dffn2 6739 . . . 4 (𝐴 Fn dom 𝐴𝐴:dom 𝐴⟶V)
3 dmcnvcnv 5947 . . . . 5 dom 𝐴 = dom 𝐴
4 df-fn 6566 . . . . 5 (𝐴 Fn dom 𝐴 ↔ (Fun 𝐴 ∧ dom 𝐴 = dom 𝐴))
53, 4mpbiran2 710 . . . 4 (𝐴 Fn dom 𝐴 ↔ Fun 𝐴)
62, 5bitr3i 277 . . 3 (𝐴:dom 𝐴⟶V ↔ Fun 𝐴)
7 relcnv 6125 . . . . 5 Rel 𝐴
8 dfrel2 6211 . . . . 5 (Rel 𝐴𝐴 = 𝐴)
97, 8mpbi 230 . . . 4 𝐴 = 𝐴
109funeqi 6589 . . 3 (Fun 𝐴 ↔ Fun 𝐴)
116, 10anbi12ci 629 . 2 ((𝐴:dom 𝐴⟶V ∧ Fun 𝐴) ↔ (Fun 𝐴 ∧ Fun 𝐴))
121, 11bitri 275 1 (𝐴:dom 𝐴1-1→V ↔ (Fun 𝐴 ∧ Fun 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1537  Vcvv 3478  ccnv 5688  dom cdm 5689  Rel wrel 5694  Fun wfun 6557   Fn wfn 6558  wf 6559  1-1wf1 6560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568
This theorem is referenced by: (None)
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