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Theorem f1cnvcnv 5411
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 5226 . 2  |-  ( `' `' A :  dom  A -1-1-> _V  <->  ( `' `' A :  dom  A --> _V  /\  Fun  `' `' `' A ) )
2 dffn2 5356 . . . 4  |-  ( `' `' A  Fn  dom  A  <->  `' `' A :  dom  A --> _V )
3 dmcnvcnv 4900 . . . . 5  |-  dom  `' `' A  =  dom  A
4 df-fn 5224 . . . . 5  |-  ( `' `' A  Fn  dom  A  <-> 
( Fun  `' `' A  /\  dom  `' `' A  =  dom  A ) )
53, 4mpbiran2 885 . . . 4  |-  ( `' `' A  Fn  dom  A  <->  Fun  `' `' A )
62, 5bitr3i 242 . . 3  |-  ( `' `' A :  dom  A --> _V 
<->  Fun  `' `' A
)
7 relcnv 5050 . . . . 5  |-  Rel  `' A
8 dfrel2 5123 . . . . 5  |-  ( Rel  `' A  <->  `' `' `' A  =  `' A )
97, 8mpbi 199 . . . 4  |-  `' `' `' A  =  `' A
109funeqi 5241 . . 3  |-  ( Fun  `' `' `' A  <->  Fun  `' A )
116, 10anbi12ci 679 . 2  |-  ( ( `' `' A :  dom  A --> _V  /\  Fun  `' `' `' A )  <->  ( Fun  `' A  /\  Fun  `' `' A ) )
121, 11bitri 240 1  |-  ( `' `' A :  dom  A -1-1-> _V  <->  ( Fun  `' A  /\  Fun  `' `' A ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623   _Vcvv 2789   `'ccnv 4687    dom cdm 4688   Rel wrel 4693   Fun wfun 5215    Fn wfn 5216   -->wf 5217   -1-1->wf1 5218
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-br 4025  df-opab 4079  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226
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