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Theorem f1cnvcnv 5348
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 4651 . 2  |-  ( `' `' A : dom  A -1-1-> _V  <->  ( `' `' A : dom  A --> _V  /\  Fun  `' `' `' A ) )
2 dffn2 5293 . . . 4  |-  ( `' `' A  Fn  dom  A  <->  `' `' A : dom  A --> _V )
3 dmcnvcnv 4854 . . . . 5  |-  dom  `' `' A  =  dom  A
4 df-fn 4649 . . . . 5  |-  ( `' `' A  Fn  dom  A  <-> 
( Fun  `' `' A  /\  dom  `' `' A  =  dom  A ) )
53, 4mpbiran2 890 . . . 4  |-  ( `' `' A  Fn  dom  A  <->  Fun  `' `' A )
62, 5bitr3i 244 . . 3  |-  ( `' `' A : dom  A --> _V 
<->  Fun  `' `' A
)
7 relcnv 5004 . . . . 5  |-  Rel  `' A
8 dfrel2 5077 . . . . 5  |-  ( Rel  `' A  <->  `' `' `' A  =  `' A )
97, 8mpbi 201 . . . 4  |-  `' `' `' A  =  `' A
109funeqi 5179 . . 3  |-  ( Fun  `' `' `' A  <->  Fun  `' A )
116, 10anbi12ci 682 . 2  |-  ( ( `' `' A : dom  A --> _V  /\  Fun  `' `' `' A )  <->  ( Fun  `' A  /\  Fun  `' `' A ) )
121, 11bitri 242 1  |-  ( `' `' A : dom  A -1-1-> _V  <->  ( Fun  `' A  /\  Fun  `' `' A ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    = wceq 1619   _Vcvv 2740   `'ccnv 4625   dom cdm 4626   Rel wrel 4631   Fun wfun 4632    Fn wfn 4633   -->wf 4634   -1-1->wf1 4635
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pr 4152
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-rab 2523  df-v 2742  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-sn 3587  df-pr 3588  df-op 3590  df-br 3964  df-opab 4018  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651
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