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Theorem f1cnvcnv 6749
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 6502 . 2 (𝐴:dom 𝐴1-1→V ↔ (𝐴:dom 𝐴⟶V ∧ Fun 𝐴))
2 dffn2 6671 . . . 4 (𝐴 Fn dom 𝐴𝐴:dom 𝐴⟶V)
3 dmcnvcnv 5889 . . . . 5 dom 𝐴 = dom 𝐴
4 df-fn 6500 . . . . 5 (𝐴 Fn dom 𝐴 ↔ (Fun 𝐴 ∧ dom 𝐴 = dom 𝐴))
53, 4mpbiran2 709 . . . 4 (𝐴 Fn dom 𝐴 ↔ Fun 𝐴)
62, 5bitr3i 277 . . 3 (𝐴:dom 𝐴⟶V ↔ Fun 𝐴)
7 relcnv 6057 . . . . 5 Rel 𝐴
8 dfrel2 6142 . . . . 5 (Rel 𝐴𝐴 = 𝐴)
97, 8mpbi 229 . . . 4 𝐴 = 𝐴
109funeqi 6523 . . 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 205  wa 397   = wceq 1542  Vcvv 3446  ccnv 5633  dom cdm 5634  Rel wrel 5639  Fun wfun 6491   Fn wfn 6492  wf 6493  1-1wf1 6494
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502
This theorem is referenced by: (None)
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