Theorem f1cnvcnv 5100

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 4907	. 2 |- ( `' `' A : dom A -1-1-> _V <-> ( `' `' A : dom A --> _V /\ Fun `' `' `' A ) )
2		dffn2 5047	. . . 4 |- ( `' `' A Fn dom A <-> `' `' A : dom A --> _V )
3		dmcnvcnv 4558	. . . . 5 |- dom `' `' A = dom A
4		df-fn 4905	. . . . 5 |- ( `' `' A Fn dom A <-> ( Fun `' `' A /\ dom `' `' A = dom A ) )
5	3, 4	mpbiran2 848	. . . 4 |- ( `' `' A Fn dom A <-> Fun `' `' A )
6	2, 5	bitr3i 175	. . 3 |- ( `' `' A : dom A --> _V <-> Fun `' `' A )
7		relcnv 4703	. . . . 5 |- Rel `' A
8		dfrel2 4771	. . . . 5 |- ( Rel `' A <-> `' `' `' A = `' A )
9	7, 8	mpbi 133	. . . 4 |- `' `' `' A = `' A
10	9	funeqi 4922	. . 3 |- ( Fun `' `' `' A <-> Fun `' A )
11	6, 10	anbi12ci 434	. 2 |- ( ( `' `' A : dom A --> _V /\ Fun `' `' `' A ) <-> ( Fun `' A /\ Fun `' `' A ) )
12	1, 11	bitri 173	1 |- ( `' `' A : dom A -1-1-> _V <-> ( Fun `' A /\ Fun `' `' A ) )

Syntax hints:   /\ wa 97   <-> wb 98   = wceq 1243   _Vcvv 2557   `'ccnv 4344   dom cdm 4345   Rel wrel 4350   Fun wfun 4896   Fn wfn 4897   -->wf 4898   -1-1->wf1 4899

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907

This theorem is referenced by: (None)