Theorem f1cnvcnv 5470

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 5259	. 2 |- ( `' `' A : dom A -1-1-> _V <-> ( `' `' A : dom A --> _V /\ Fun `' `' `' A ) )
2		dffn2 5405	. . . 4 |- ( `' `' A Fn dom A <-> `' `' A : dom A --> _V )
3		dmcnvcnv 4886	. . . . 5 |- dom `' `' A = dom A
4		df-fn 5257	. . . . 5 |- ( `' `' A Fn dom A <-> ( Fun `' `' A /\ dom `' `' A = dom A ) )
5	3, 4	mpbiran2 943	. . . 4 |- ( `' `' A Fn dom A <-> Fun `' `' A )
6	2, 5	bitr3i 186	. . 3 |- ( `' `' A : dom A --> _V <-> Fun `' `' A )
7		relcnv 5043	. . . . 5 |- Rel `' A
8		dfrel2 5116	. . . . 5 |- ( Rel `' A <-> `' `' `' A = `' A )
9	7, 8	mpbi 145	. . . 4 |- `' `' `' A = `' A
10	9	funeqi 5275	. . 3 |- ( Fun `' `' `' A <-> Fun `' A )
11	6, 10	anbi12ci 461	. 2 |- ( ( `' `' A : dom A --> _V /\ Fun `' `' `' A ) <-> ( Fun `' A /\ Fun `' `' A ) )
12	1, 11	bitri 184	1 |- ( `' `' A : dom A -1-1-> _V <-> ( Fun `' A /\ Fun `' `' A ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1364   _Vcvv 2760   `'ccnv 4658   dom cdm 4659   Rel wrel 4664   Fun wfun 5248   Fn wfn 5249   -->wf 5250   -1-1->wf1 5251

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259

This theorem is referenced by: (None)