Theorem f1funfun 5264

Description: Two ways to express that a set A 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 set.mm contributors, 13-Aug-2004.) (Revised by Scott Fenton, 18-Apr-2021.)

Assertion

Ref	Expression
f1funfun	|- (A:dom A-1-1->_V <-> (Fun `'A /\ Fun A))

Proof of Theorem f1funfun

Step	Hyp	Ref	Expression
1		df-f1 4793	. 2 |- (A:dom A-1-1->_V <-> (A:dom A-->_V /\ Fun `'A))
2		ancom 437	. 2 |- ((A:dom A-->_V /\ Fun `'A) <-> (Fun `'A /\ A:dom A-->_V))
3		ssv 3292	. . . . 5 |- ran A C_ _V
4		df-f 4792	. . . . 5 |- (A:dom A-->_V <-> (A Fn dom A /\ ran A C_ _V))
5	3, 4	mpbiran2 885	. . . 4 |- (A:dom A-->_V <-> A Fn dom A)
6		funfn 5137	. . . 4 |- (Fun A <-> A Fn dom A)
7	5, 6	bitr4i 243	. . 3 |- (A:dom A-->_V <-> Fun A)
8	7	anbi2i 675	. 2 |- ((Fun `'A /\ A:dom A-->_V) <-> (Fun `'A /\ Fun A))
9	1, 2, 8	3bitri 262	1 |- (A:dom A-1-1->_V <-> (Fun `'A /\ Fun A))

Syntax hints:   <-> wb 176   /\ wa 358  _Vcvv 2860   C_ wss 3258  `'ccnv 4772  dom cdm 4773  ran crn 4774  Fun wfun 4776   Fn wfn 4777  -->wf 4778  -1-1->wf1 4779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-fn 4791  df-f 4792  df-f1 4793

This theorem is referenced by: (None)