Theorem f1cnvcnv 5562

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 5338	. 2 |- ( `' `' A : dom A -1-1-> _V <-> ( `' `' A : dom A --> _V /\ Fun `' `' `' A ) )
2		dffn2 5491	. . . 4 |- ( `' `' A Fn dom A <-> `' `' A : dom A --> _V )
3		dmcnvcnv 4962	. . . . 5 |- dom `' `' A = dom A
4		df-fn 5336	. . . . 5 |- ( `' `' A Fn dom A <-> ( Fun `' `' A /\ dom `' `' A = dom A ) )
5	3, 4	mpbiran2 950	. . . 4 |- ( `' `' A Fn dom A <-> Fun `' `' A )
6	2, 5	bitr3i 186	. . 3 |- ( `' `' A : dom A --> _V <-> Fun `' `' A )
7		relcnv 5121	. . . . 5 |- Rel `' A
8		dfrel2 5194	. . . . 5 |- ( Rel `' A <-> `' `' `' A = `' A )
9	7, 8	mpbi 145	. . . 4 |- `' `' `' A = `' A
10	9	funeqi 5354	. . 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 1398   _Vcvv 2803   `'ccnv 4730   dom cdm 4731   Rel wrel 4736   Fun wfun 5327   Fn wfn 5328   -->wf 5329   -1-1->wf1 5330

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338

This theorem is referenced by: (None)