Theorem f1cnv 4611

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.

Assertion

Ref	Expression
f1cnv	|- (`'`'A:dom A-1-1->_V <-> (Fun `'A /\ Fun `'`'A))

Proof of Theorem f1cnv

Step	Hyp	Ref	Expression
1		df-f1 4011	. 2 |- (`'`'A:dom A-1-1->_V <-> (`'`'A:dom A-->_V /\ Fun `'`'`'A))
2		dffn2 4563	. . . 4 |- (`'`'A Fn dom A <-> `'`'A:dom A-->_V)
3		df-fn 4009	. . . . 5 |- (`'`'A Fn dom A <-> (Fun `'`'A /\ dom `'`'A = dom A))
4		dmcnvcnv 4183	. . . . 5 |- dom `'`'A = dom A
5	3, 4	mpbiran2 799	. . . 4 |- (`'`'A Fn dom A <-> Fun `'`'A)
6	2, 5	bitr3i 192	. . 3 |- (`'`'A:dom A-->_V <-> Fun `'`'A)
7		relcnv 4301	. . . . 5 |- Rel `'A
8		dfrel2 4358	. . . . 5 |- (Rel `'A <-> `'`'`'A = `'A)
9	7, 8	mpbi 206	. . . 4 |- `'`'`'A = `'A
10		funeq 4441	. . . 4 |- (`'`'`'A = `'A -> (Fun `'`'`'A <-> Fun `'A))
11	9, 10	ax-mp 7	. . 3 |- (Fun `'`'`'A <-> Fun `'A)
12	6, 11	anbi12i 540	. 2 |- ((`'`'A:dom A-->_V /\ Fun `'`'`'A) <-> (Fun `'`'A /\ Fun `'A))
13		ancom 482	. 2 |- ((Fun `'`'A /\ Fun `'A) <-> (Fun `'A /\ Fun `'`'A))
14	1, 12, 13	3bitri 194	1 |- (`'`'A:dom A-1-1->_V <-> (Fun `'A /\ Fun `'`'A))

Syntax hints:   <-> wb 163   /\ wa 240   = wceq 1298  _Vcvv 2292  `'ccnv 3985  dom cdm 3986  Rel wrel 3991  Fun wfun 3992   Fn wfn 3993  -->wf 3994  -1-1->wf1 3995

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011

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