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Theorem f1cnvcnv 5447
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 5262 . 2  |-  ( `' `' A : dom  A -1-1-> _V  <->  ( `' `' A : dom  A --> _V  /\  Fun  `' `' `' A ) )
2 dffn2 5392 . . . 4  |-  ( `' `' A  Fn  dom  A  <->  `' `' A : dom  A --> _V )
3 dmcnvcnv 4903 . . . . 5  |-  dom  `' `' A  =  dom  A
4 df-fn 5260 . . . . 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 5053 . . . . 5  |-  Rel  `' A
8 dfrel2 5126 . . . . 5  |-  ( Rel  `' A  <->  `' `' `' A  =  `' A )
97, 8mpbi 199 . . . 4  |-  `' `' `' A  =  `' A
109funeqi 5277 . . 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 1625   _Vcvv 2790   `'ccnv 4690   dom cdm 4691   Rel wrel 4696   Fun wfun 5251    Fn wfn 5252   -->wf 5253   -1-1->wf1 5254
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4216
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-br 4026  df-opab 4080  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262
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