Theorem f1cnvcnv 5221

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 5015	. 2 |- ( `' `' A : dom A -1-1-> _V <-> ( `' `' A : dom A --> _V /\ Fun `' `' `' A ) )
2		dffn2 5157	. . . 4 |- ( `' `' A Fn dom A <-> `' `' A : dom A --> _V )
3		dmcnvcnv 4655	. . . . 5 |- dom `' `' A = dom A
4		df-fn 5013	. . . . 5 |- ( `' `' A Fn dom A <-> ( Fun `' `' A /\ dom `' `' A = dom A ) )
5	3, 4	mpbiran2 887	. . . 4 |- ( `' `' A Fn dom A <-> Fun `' `' A )
6	2, 5	bitr3i 184	. . 3 |- ( `' `' A : dom A --> _V <-> Fun `' `' A )
7		relcnv 4805	. . . . 5 |- Rel `' A
8		dfrel2 4876	. . . . 5 |- ( Rel `' A <-> `' `' `' A = `' A )
9	7, 8	mpbi 143	. . . 4 |- `' `' `' A = `' A
10	9	funeqi 5030	. . 3 |- ( Fun `' `' `' A <-> Fun `' A )
11	6, 10	anbi12ci 449	. 2 |- ( ( `' `' A : dom A --> _V /\ Fun `' `' `' A ) <-> ( Fun `' A /\ Fun `' `' A ) )
12	1, 11	bitri 182	1 |- ( `' `' A : dom A -1-1-> _V <-> ( Fun `' A /\ Fun `' `' A ) )

Syntax hints:   /\ wa 102   <-> wb 103   = wceq 1289   _Vcvv 2619   `'ccnv 4435   dom cdm 4436   Rel wrel 4441   Fun wfun 5004   Fn wfn 5005   -->wf 5006   -1-1->wf1 5007

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3955  ax-pow 4007  ax-pr 4034

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453  df-br 3844  df-opab 3898  df-xp 4442  df-rel 4443  df-cnv 4444  df-co 4445  df-dm 4446  df-rn 4447  df-fun 5012  df-fn 5013  df-f 5014  df-f1 5015

This theorem is referenced by: (None)