Theorem f1cnvcnv 5794

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 5598	. 2
2		dffn2 5737	. . . 4
3		dmcnvcnv 5230	. . . . 5
4		df-fn 5596	. . . . 5
5	3, 4	mpbiran2 919	. . . 4
6	2, 5	bitr3i 251	. . 3
7		relcnv 5379	. . . . 5
8		dfrel2 5462	. . . . 5
9	7, 8	mpbi 208	. . . 4
10	9	funeqi 5613	. . 3
11	6, 10	anbi12ci 698	. 2
12	1, 11	bitri 249	1

Syntax hints:  <->wb 184  /\wa 369  =wceq 1395  cvv 3109  `'ccnv 5003  domcdm 5004  Relwrel 5009  Funwfun 5587  Fnwfn 5588  -->wf 5589  -1-1->wf1 5590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453  df-opab 4511  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598