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Theorem f1funfun 5263
 Description: Two ways to express that a set A 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 set.mm contributors, 13-Aug-2004.) (Revised by Scott Fenton, 18-Apr-2021.)
Assertion
Ref Expression
f1funfun (A:dom A1-1→V ↔ (Fun A Fun A))

Proof of Theorem f1funfun
StepHypRef Expression
1 df-f1 4792 . 2 (A:dom A1-1→V ↔ (A:dom A–→V Fun A))
2 ancom 437 . 2 ((A:dom A–→V Fun A) ↔ (Fun A A:dom A–→V))
3 ssv 3291 . . . . 5 ran A V
4 df-f 4791 . . . . 5 (A:dom A–→V ↔ (A Fn dom A ran A V))
53, 4mpbiran2 885 . . . 4 (A:dom A–→V ↔ A Fn dom A)
6 funfn 5136 . . . 4 (Fun AA Fn dom A)
75, 6bitr4i 243 . . 3 (A:dom A–→V ↔ Fun A)
87anbi2i 675 . 2 ((Fun A A:dom A–→V) ↔ (Fun A Fun A))
91, 2, 83bitri 262 1 (A:dom A1-1→V ↔ (Fun A Fun A))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358  Vcvv 2859   ⊆ wss 3257  ◡ccnv 4771  dom cdm 4772  ran crn 4773  Fun wfun 4775   Fn wfn 4776  –→wf 4777  –1-1→wf1 4778 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-fn 4790  df-f 4791  df-f1 4792 This theorem is referenced by: (None)
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