Theorem f1cnvcnv 5347

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

Step	Hyp	Ref	Expression
1		df-f1 5136	. 2 |- ( `' `' A : dom A -1-1-> _V <-> ( `' `' A : dom A --> _V /\ Fun `' `' `' A ) )
2		dffn2 5282	. . . 4 |- ( `' `' A Fn dom A <-> `' `' A : dom A --> _V )
3		dmcnvcnv 4771	. . . . 5 |- dom `' `' A = dom A
4		df-fn 5134	. . . . 5 |- ( `' `' A Fn dom A <-> ( Fun `' `' A /\ dom `' `' A = dom A ) )
5	3, 4	mpbiran2 926	. . . 4 |- ( `' `' A Fn dom A <-> Fun `' `' A )
6	2, 5	bitr3i 185	. . 3 |- ( `' `' A : dom A --> _V <-> Fun `' `' A )
7		relcnv 4925	. . . . 5 |- Rel `' A
8		dfrel2 4997	. . . . 5 |- ( Rel `' A <-> `' `' `' A = `' A )
9	7, 8	mpbi 144	. . . 4 |- `' `' `' A = `' A
10	9	funeqi 5152	. . 3 |- ( Fun `' `' `' A <-> Fun `' A )
11	6, 10	anbi12ci 457	. 2 |- ( ( `' `' A : dom A --> _V /\ Fun `' `' `' A ) <-> ( Fun `' A /\ Fun `' `' A ) )
12	1, 11	bitri 183	1 |- ( `' `' A : dom A -1-1-> _V <-> ( Fun `' A /\ Fun `' `' A ) )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1332   _Vcvv 2689   `'ccnv 4546   dom cdm 4547   Rel wrel 4552   Fun wfun 5125   Fn wfn 5126   -->wf 5127   -1-1->wf1 5128

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136

This theorem is referenced by: (None)