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Theorem bj-charfunr 13345

Description: If a class

has a "weak" characteristic function on a class

, then negated membership in

is decidable (in other words, membership in

is testable) in

The hypothesis imposes that be a set. As usual, it could be formulated as $/\$ to deal with general classes, but that extra generality would not make the theorem much more useful.

The theorem would still hold if the codomain of were any class with testable equality to the point where $\$ is sent. (Contributed by BJ, 6-Aug-2024.)

**Hypothesis**
Ref	Expression
bj-charfunr.1	$/\$ $\$

**Assertion**
Ref	Expression
bj-charfunr	DECID

Distinct variable groups:

Allowed substitution hints:

(

)

(

)

**Proof of Theorem bj-charfunr**
Step	Hyp	Ref	Expression
1		bj-charfunr.1	. . . . 5 $/\$ $\$
2		elmapi 6608	. . . . . . . . . 10
3		ffvelrn 5597	. . . . . . . . . . 11 $/\$
4	3	ex 114	. . . . . . . . . 10
5	2, 4	syl 14	. . . . . . . . 9
6		0elnn 4576	. . . . . . . . . 10 $\/$
7		nn0eln0 4577	. . . . . . . . . . 11
8	7	orbi2d 780	. . . . . . . . . 10 $\/$ $\/$
9	6, 8	mpbid 146	. . . . . . . . 9 $\/$
10	5, 9	syl6 33	. . . . . . . 8 $\/$
11	10	adantr 274	. . . . . . 7 $/\$ $/\$ $\$ $\/$
12		elin 3290	. . . . . . . . . . . . . . 15 $/\$
13		rsp 2504	. . . . . . . . . . . . . . 15
14	12, 13	syl5bir 152	. . . . . . . . . . . . . 14 $/\$
15	14	expd 256	. . . . . . . . . . . . 13
16	15	adantr 274	. . . . . . . . . . . 12 $/\$ $\$
17	16	imp 123	. . . . . . . . . . 11 $/\$ $\$ $/\$
18	17	necon2bd 2385	. . . . . . . . . 10 $/\$ $\$ $/\$
19		eldif 3111	. . . . . . . . . . . . . . 15 $\$ $/\$
20		rsp 2504	. . . . . . . . . . . . . . 15 $\$ $\$
21	19, 20	syl5bir 152	. . . . . . . . . . . . . 14 $\$ $/\$
22	21	expd 256	. . . . . . . . . . . . 13 $\$
23	22	adantl 275	. . . . . . . . . . . 12 $/\$ $\$
24	23	imp 123	. . . . . . . . . . 11 $/\$ $\$ $/\$
25	24	necon3ad 2369	. . . . . . . . . 10 $/\$ $\$ $/\$
26	18, 25	orim12d 776	. . . . . . . . 9 $/\$ $\$ $/\$ $\/$ $\/$
27	26	ex 114	. . . . . . . 8 $/\$ $\$ $\/$ $\/$
28	27	adantl 275	. . . . . . 7 $/\$ $/\$ $\$ $\/$ $\/$
29	11, 28	mpdd 41	. . . . . 6 $/\$ $/\$ $\$ $\/$
30	29	adantl 275	. . . . 5 $/\$ $/\$ $/\$ $\$ $\/$
31	1, 30	rexlimddv 2579	. . . 4 $\/$
32	31	imp 123	. . 3 $/\$ $\/$
33		df-dc 821	. . 3 DECID $\/$
34	32, 33	sylibr 133	. 2 $/\$ DECID
35	34	ralrimiva 2530	1 DECID

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 103 $\/$ wo 698 DECID wdc 820

wceq 1335

wcel 2128

wne 2327

wral 2435

wrex 2436 $\$ cdif 3099

cin 3101

c0 3394

com 4547

wf 5163

cfv 5167 (class class class)co 5818

cmap 6586

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-nul 4090 ax-pow 4134 ax-pr 4168 ax-un 4392 ax-setind 4494 ax-iinf 4545

This theorem depends on definitions: df-bi 116 df-dc 821 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-br 3966 df-opab 4026 df-id 4252 df-suc 4330 df-iom 4548 df-xp 4589 df-rel 4590 df-cnv 4591 df-co 4592 df-dm 4593 df-rn 4594 df-iota 5132 df-fun 5169 df-fn 5170 df-f 5171 df-fv 5175 df-ov 5821 df-oprab 5822 df-mpo 5823 df-map 6588

This theorem is referenced by: bj-charfunbi 13346

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