Theorem f1cnv 2782

Description: Two ways of expressing 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 do not use it very often.

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 2435	. 2 |- (`'`'A:dom A-1-1->V <-> (`'`'A:dom A-->V /\ Fun `'`'`'A))
2		fnf 2753	. . . 4 |- (`'`'A Fn dom A <-> `'`'A:dom A-->V)
3		df-fn 2433	. . . . 5 |- (`'`'A Fn dom A <-> (Fun `'`'A /\ dom `'`'A = dom A))
4		dfdm4 2525	. . . . . 6 |- dom A = ran `'A
5		df-rn 2429	. . . . . 6 |- ran `'A = dom `'`'A
6	4, 5	eqtr2 1120	. . . . 5 |- dom `'`'A = dom A
7	3, 6	mpbiranr 548	. . . 4 |- (`'`'A Fn dom A <-> Fun `'`'A)
8	2, 7	bitr3 153	. . 3 |- (`'`'A:dom A-->V <-> Fun `'`'A)
9		relcnv 2624	. . . . 5 |- Rel `'A
10		dfrel2 2660	. . . . 5 |- (Rel `'A <-> `'`'`'A = `'A)
11	9, 10	mpbi 164	. . . 4 |- `'`'`'A = `'A
12		funeq 2683	. . . 4 |- (`'`'`'A = `'A -> (Fun `'`'`'A <-> Fun `'A))
13	11, 12	ax-mp 6	. . 3 |- (Fun `'`'`'A <-> Fun `'A)
14	8, 13	anbi12i 369	. 2 |- ((`'`'A:dom A-->V /\ Fun `'`'`'A) <-> (Fun `'`'A /\ Fun `'A))
15		ancom 333	. 2 |- ((Fun `'`'A /\ Fun `'A) <-> (Fun `'A /\ Fun `'`'A))
16	1, 14, 15	3bitr 155	1 |- (`'`'A:dom A-1-1->V <-> (Fun `'A /\ Fun `'`'A))

Syntax hints:   <-> wb 127   /\ wa 196   = wceq 1091  Vcvv 1348  `'ccnv 2409  dom cdm 2410  ran crn 2411  Rel wrel 2415  Fun wfun 2416   Fn wfn 2417  -->wf 2418  -1-1->wf1 2419

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435

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