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Theorem f1cnvcnv 6022
 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 5809 . 2 (𝐴:dom 𝐴1-1→V ↔ (𝐴:dom 𝐴⟶V ∧ Fun 𝐴))
2 dffn2 5960 . . . 4 (𝐴 Fn dom 𝐴𝐴:dom 𝐴⟶V)
3 dmcnvcnv 5269 . . . . 5 dom 𝐴 = dom 𝐴
4 df-fn 5807 . . . . 5 (𝐴 Fn dom 𝐴 ↔ (Fun 𝐴 ∧ dom 𝐴 = dom 𝐴))
53, 4mpbiran2 956 . . . 4 (𝐴 Fn dom 𝐴 ↔ Fun 𝐴)
62, 5bitr3i 265 . . 3 (𝐴:dom 𝐴⟶V ↔ Fun 𝐴)
7 relcnv 5422 . . . . 5 Rel 𝐴
8 dfrel2 5502 . . . . 5 (Rel 𝐴𝐴 = 𝐴)
97, 8mpbi 219 . . . 4 𝐴 = 𝐴
109funeqi 5824 . . 3 (Fun 𝐴 ↔ Fun 𝐴)
116, 10anbi12ci 730 . 2 ((𝐴:dom 𝐴⟶V ∧ Fun 𝐴) ↔ (Fun 𝐴 ∧ Fun 𝐴))
121, 11bitri 263 1 (𝐴:dom 𝐴1-1→V ↔ (Fun 𝐴 ∧ Fun 𝐴))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475  Vcvv 3173  ◡ccnv 5037  dom cdm 5038  Rel wrel 5043  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  –1-1→wf1 5801 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809 This theorem is referenced by: (None)
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