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Theorem bj-charfunr 16173

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 6817	. . . . . . . . . 10
3		ffvelcdm 5768	. . . . . . . . . . 11 $/\$
4	3	ex 115	. . . . . . . . . 10
5	2, 4	syl 14	. . . . . . . . 9
6		0elnn 4711	. . . . . . . . . 10 $\/$
7		nn0eln0 4712	. . . . . . . . . . 11
8	7	orbi2d 795	. . . . . . . . . 10 $\/$ $\/$
9	6, 8	mpbid 147	. . . . . . . . 9 $\/$
10	5, 9	syl6 33	. . . . . . . 8 $\/$
11	10	adantr 276	. . . . . . 7 $/\$ $/\$ $\$ $\/$
12		elin 3387	. . . . . . . . . . . . . . 15 $/\$
13		rsp 2577	. . . . . . . . . . . . . . 15
14	12, 13	biimtrrid 153	. . . . . . . . . . . . . 14 $/\$
15	14	expd 258	. . . . . . . . . . . . 13
16	15	adantr 276	. . . . . . . . . . . 12 $/\$ $\$
17	16	imp 124	. . . . . . . . . . 11 $/\$ $\$ $/\$
18	17	necon2bd 2458	. . . . . . . . . 10 $/\$ $\$ $/\$
19		eldif 3206	. . . . . . . . . . . . . . 15 $\$ $/\$
20		rsp 2577	. . . . . . . . . . . . . . 15 $\$ $\$
21	19, 20	biimtrrid 153	. . . . . . . . . . . . . 14 $\$ $/\$
22	21	expd 258	. . . . . . . . . . . . 13 $\$
23	22	adantl 277	. . . . . . . . . . . 12 $/\$ $\$
24	23	imp 124	. . . . . . . . . . 11 $/\$ $\$ $/\$
25	24	necon3ad 2442	. . . . . . . . . 10 $/\$ $\$ $/\$
26	18, 25	orim12d 791	. . . . . . . . 9 $/\$ $\$ $/\$ $\/$ $\/$
27	26	ex 115	. . . . . . . 8 $/\$ $\$ $\/$ $\/$
28	27	adantl 277	. . . . . . 7 $/\$ $/\$ $\$ $\/$ $\/$
29	11, 28	mpdd 41	. . . . . 6 $/\$ $/\$ $\$ $\/$
30	29	adantl 277	. . . . 5 $/\$ $/\$ $/\$ $\$ $\/$
31	1, 30	rexlimddv 2653	. . . 4 $\/$
32	31	imp 124	. . 3 $/\$ $\/$
33		df-dc 840	. . 3 DECID $\/$
34	32, 33	sylibr 134	. 2 $/\$ DECID
35	34	ralrimiva 2603	1 DECID

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104 $\/$ wo 713 DECID wdc 839

wceq 1395

wcel 2200

wne 2400

wral 2508

wrex 2509 $\$ cdif 3194

cin 3196

c0 3491

com 4682

wf 5314

cfv 5318 (class class class)co 6001

cmap 6795

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-sep 4202 ax-nul 4210 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-iinf 4680

This theorem depends on definitions: df-bi 117 df-dc 840 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-ral 2513 df-rex 2514 df-v 2801 df-sbc 3029 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-br 4084 df-opab 4146 df-id 4384 df-suc 4462 df-iom 4683 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-iota 5278 df-fun 5320 df-fn 5321 df-f 5322 df-fv 5326 df-ov 6004 df-oprab 6005 df-mpo 6006 df-map 6797

This theorem is referenced by: bj-charfunbi 16174

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