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Theorem f1cnvcnv 5427
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 5216 . 2  |-  ( `' `' A : dom  A -1-1-> _V  <->  ( `' `' A : dom  A --> _V  /\  Fun  `' `' `' A ) )
2 dffn2 5362 . . . 4  |-  ( `' `' A  Fn  dom  A  <->  `' `' A : dom  A --> _V )
3 dmcnvcnv 4846 . . . . 5  |-  dom  `' `' A  =  dom  A
4 df-fn 5214 . . . . 5  |-  ( `' `' A  Fn  dom  A  <-> 
( Fun  `' `' A  /\  dom  `' `' A  =  dom  A ) )
53, 4mpbiran2 941 . . . 4  |-  ( `' `' A  Fn  dom  A  <->  Fun  `' `' A )
62, 5bitr3i 186 . . 3  |-  ( `' `' A : dom  A --> _V 
<->  Fun  `' `' A
)
7 relcnv 5001 . . . . 5  |-  Rel  `' A
8 dfrel2 5074 . . . . 5  |-  ( Rel  `' A  <->  `' `' `' A  =  `' A )
97, 8mpbi 145 . . . 4  |-  `' `' `' A  =  `' A
109funeqi 5232 . . 3  |-  ( Fun  `' `' `' A  <->  Fun  `' A )
116, 10anbi12ci 461 . 2  |-  ( ( `' `' A : dom  A --> _V  /\  Fun  `' `' `' A )  <->  ( Fun  `' A  /\  Fun  `' `' A ) )
121, 11bitri 184 1  |-  ( `' `' A : dom  A -1-1-> _V  <->  ( Fun  `' A  /\  Fun  `' `' A ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105    = wceq 1353   _Vcvv 2737   `'ccnv 4621   dom cdm 4622   Rel wrel 4627   Fun wfun 5205    Fn wfn 5206   -->wf 5207   -1-1->wf1 5208
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-br 4001  df-opab 4062  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216
This theorem is referenced by: (None)
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