Theorem f1cnvcnv 5694

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 5506	. 2
2		dffn2 5639	. . . 4
3		dmcnvcnv 5136	. . . . 5
4		df-fn 5504	. . . . 5
5	3, 4	mpbiran2 887	. . . 4
6	2, 5	bitr3i 244	. . 3
7		relcnv 5286	. . . . 5
8		dfrel2 5365	. . . . 5
9	7, 8	mpbi 201	. . . 4
10	9	funeqi 5521	. . 3
11	6, 10	anbi12ci 681	. 2
12	1, 11	bitri 242	1

Syntax hints:  <->wb 178  /\wa 360  =wceq 1654  cvv 2965  `'ccnv 4918  domcdm 4919  Relwrel 4924  Funwfun 5495  Fnwfn 5496  -->wf 5497  -1-1->wf1 5498

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-sep 4364  ax-nul 4372  ax-pr 4442

This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2717  df-rex 2718  df-rab 2721  df-v 2967  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-nul 3617  df-if 3766  df-sn 3847  df-pr 3848  df-op 3850  df-br 4244  df-opab 4302  df-xp 4925  df-rel 4926  df-cnv 4927  df-co 4928  df-dm 4929  df-rn 4930  df-fun 5503  df-fn 5504  df-f 5505  df-f1 5506