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Theorem f1cnvcnv 6066
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 5852 . 2 (𝐴:dom 𝐴1-1→V ↔ (𝐴:dom 𝐴⟶V ∧ Fun 𝐴))
2 dffn2 6004 . . . 4 (𝐴 Fn dom 𝐴𝐴:dom 𝐴⟶V)
3 dmcnvcnv 5308 . . . . 5 dom 𝐴 = dom 𝐴
4 df-fn 5850 . . . . 5 (𝐴 Fn dom 𝐴 ↔ (Fun 𝐴 ∧ dom 𝐴 = dom 𝐴))
53, 4mpbiran2 953 . . . 4 (𝐴 Fn dom 𝐴 ↔ Fun 𝐴)
62, 5bitr3i 266 . . 3 (𝐴:dom 𝐴⟶V ↔ Fun 𝐴)
7 relcnv 5462 . . . . 5 Rel 𝐴
8 dfrel2 5542 . . . . 5 (Rel 𝐴𝐴 = 𝐴)
97, 8mpbi 220 . . . 4 𝐴 = 𝐴
109funeqi 5868 . . 3 (Fun 𝐴 ↔ Fun 𝐴)
116, 10anbi12ci 733 . 2 ((𝐴:dom 𝐴⟶V ∧ Fun 𝐴) ↔ (Fun 𝐴 ∧ Fun 𝐴))
121, 11bitri 264 1 (𝐴:dom 𝐴1-1→V ↔ (Fun 𝐴 ∧ Fun 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  Vcvv 3186  ccnv 5073  dom cdm 5074  Rel wrel 5079  Fun wfun 5841   Fn wfn 5842  wf 5843  1-1wf1 5844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852
This theorem is referenced by: (None)
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