Theorem f1cnvcnv 5608

Description: Two ways to express that a set (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

Proof of Theorem f1cnvcnv

Step	Hyp	Ref	Expression
1		df-f1 5422	. 2
2		dffn2 5557	. . . 4
3		dmcnvcnv 5066	. . . . 5
4		df-fn 5420	. . . . 5
5	3, 4	mpbiran2 887	. . . 4
6	2, 5	bitr3i 244	. . 3
7		relcnv 5208	. . . . 5
8		dfrel2 5287	. . . . 5
9	7, 8	mpbi 201	. . . 4
10	9	funeqi 5437	. . 3
11	6, 10	anbi12ci 681	. 2
12	1, 11	bitri 242	1

Syntax hints:  <->wb 178  /\wa 360  =wceq 1662  cvv 3006  `'ccnv 4843  domcdm 4844  Relwrel 4849  Funwfun 5411  Fnwfn 5412  -->wf 5413  -1-1->wf1 5414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1562  ax-4 1573  ax-5 1636  ax-6 1677  ax-7 1697  ax-9 1728  ax-10 1743  ax-11 1748  ax-12 1760  ax-13 1947  ax-ext 2462  ax-sep 4423  ax-nul 4431  ax-pr 4538

This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1337  df-ex 1558  df-nf 1561  df-sb 1669  df-eu 2309  df-mo 2310  df-clab 2468  df-cleq 2474  df-clel 2477  df-nfc 2606  df-ne 2646  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3008  df-dif 3356  df-un 3358  df-in 3360  df-ss 3367  df-nul 3661  df-if 3813  df-sn 3900  df-pr 3901  df-op 3903  df-br 4303  df-opab 4361  df-xp 4850  df-rel 4851  df-cnv 4852  df-co 4853  df-dm 4854  df-rn 4855  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422