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Theorem f1cnvcnv 5787
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 5591 . 2  |-  ( `' `' A : dom  A -1-1-> _V  <->  ( `' `' A : dom  A --> _V  /\  Fun  `' `' `' A ) )
2 dffn2 5730 . . . 4  |-  ( `' `' A  Fn  dom  A  <->  `' `' A : dom  A --> _V )
3 dmcnvcnv 5223 . . . . 5  |-  dom  `' `' A  =  dom  A
4 df-fn 5589 . . . . 5  |-  ( `' `' A  Fn  dom  A  <-> 
( Fun  `' `' A  /\  dom  `' `' A  =  dom  A ) )
53, 4mpbiran2 917 . . . 4  |-  ( `' `' A  Fn  dom  A  <->  Fun  `' `' A )
62, 5bitr3i 251 . . 3  |-  ( `' `' A : dom  A --> _V 
<->  Fun  `' `' A
)
7 relcnv 5372 . . . . 5  |-  Rel  `' A
8 dfrel2 5455 . . . . 5  |-  ( Rel  `' A  <->  `' `' `' A  =  `' A )
97, 8mpbi 208 . . . 4  |-  `' `' `' A  =  `' A
109funeqi 5606 . . 3  |-  ( Fun  `' `' `' A  <->  Fun  `' A )
116, 10anbi12ci 698 . 2  |-  ( ( `' `' A : dom  A --> _V  /\  Fun  `' `' `' A )  <->  ( Fun  `' A  /\  Fun  `' `' A ) )
121, 11bitri 249 1  |-  ( `' `' A : dom  A -1-1-> _V  <->  ( Fun  `' A  /\  Fun  `' `' A ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1379   _Vcvv 3113   `'ccnv 4998   dom cdm 4999   Rel wrel 5004   Fun wfun 5580    Fn wfn 5581   -->wf 5582   -1-1->wf1 5583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591
This theorem is referenced by: (None)
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