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Theorem f1cnvcnv 5640
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  A -1-1-> _V  <->  ( Fun  `' A  /\  Fun  `' `' A ) )

Proof of Theorem f1cnvcnv
StepHypRef Expression
1 df-f1 5452 . 2  |-  ( `' `' A : dom  A -1-1-> _V  <->  ( `' `' A : dom  A --> _V  /\  Fun  `' `' `' A ) )
2 dffn2 5585 . . . 4  |-  ( `' `' A  Fn  dom  A  <->  `' `' A : dom  A --> _V )
3 dmcnvcnv 5085 . . . . 5  |-  dom  `' `' A  =  dom  A
4 df-fn 5450 . . . . 5  |-  ( `' `' A  Fn  dom  A  <-> 
( Fun  `' `' A  /\  dom  `' `' A  =  dom  A ) )
53, 4mpbiran2 886 . . . 4  |-  ( `' `' A  Fn  dom  A  <->  Fun  `' `' A )
62, 5bitr3i 243 . . 3  |-  ( `' `' A : dom  A --> _V 
<->  Fun  `' `' A
)
7 relcnv 5235 . . . . 5  |-  Rel  `' A
8 dfrel2 5314 . . . . 5  |-  ( Rel  `' A  <->  `' `' `' A  =  `' A )
97, 8mpbi 200 . . . 4  |-  `' `' `' A  =  `' A
109funeqi 5467 . . 3  |-  ( Fun  `' `' `' A  <->  Fun  `' A )
116, 10anbi12ci 680 . 2  |-  ( ( `' `' A : dom  A --> _V  /\  Fun  `' `' `' A )  <->  ( Fun  `' A  /\  Fun  `' `' A ) )
121, 11bitri 241 1  |-  ( `' `' A : dom  A -1-1-> _V  <->  ( Fun  `' A  /\  Fun  `' `' A ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1652   _Vcvv 2949   `'ccnv 4870   dom cdm 4871   Rel wrel 4876   Fun wfun 5441    Fn wfn 5442   -->wf 5443   -1-1->wf1 5444
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pr 4396
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-op 3816  df-br 4206  df-opab 4260  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452
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