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Theorem bj-charfunr 15101

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 6704	. . . . . . . . . 10
3		ffvelcdm 5677	. . . . . . . . . . 11 $/\$
4	3	ex 115	. . . . . . . . . 10
5	2, 4	syl 14	. . . . . . . . 9
6		0elnn 4643	. . . . . . . . . 10 $\/$
7		nn0eln0 4644	. . . . . . . . . . 11
8	7	orbi2d 791	. . . . . . . . . 10 $\/$ $\/$
9	6, 8	mpbid 147	. . . . . . . . 9 $\/$
10	5, 9	syl6 33	. . . . . . . 8 $\/$
11	10	adantr 276	. . . . . . 7 $/\$ $/\$ $\$ $\/$
12		elin 3337	. . . . . . . . . . . . . . 15 $/\$
13		rsp 2537	. . . . . . . . . . . . . . 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 2418	. . . . . . . . . 10 $/\$ $\$ $/\$
19		eldif 3157	. . . . . . . . . . . . . . 15 $\$ $/\$
20		rsp 2537	. . . . . . . . . . . . . . 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 2402	. . . . . . . . . 10 $/\$ $\$ $/\$
26	18, 25	orim12d 787	. . . . . . . . 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 2612	. . . 4 $\/$
32	31	imp 124	. . 3 $/\$ $\/$
33		df-dc 836	. . 3 DECID $\/$
34	32, 33	sylibr 134	. 2 $/\$ DECID
35	34	ralrimiva 2563	1 DECID

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104 $\/$ wo 709 DECID wdc 835

wceq 1364

wcel 2160

wne 2360

wral 2468

wrex 2469 $\$ cdif 3145

cin 3147

c0 3441

com 4614

wf 5238

cfv 5242 (class class class)co 5904

cmap 6682

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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4143 ax-nul 4151 ax-pow 4199 ax-pr 4234 ax-un 4458 ax-setind 4561 ax-iinf 4612

This theorem depends on definitions: df-bi 117 df-dc 836 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-v 2758 df-sbc 2982 df-dif 3150 df-un 3152 df-in 3154 df-ss 3161 df-nul 3442 df-pw 3599 df-sn 3620 df-pr 3621 df-op 3623 df-uni 3832 df-int 3867 df-br 4026 df-opab 4087 df-id 4318 df-suc 4396 df-iom 4615 df-xp 4657 df-rel 4658 df-cnv 4659 df-co 4660 df-dm 4661 df-rn 4662 df-iota 5203 df-fun 5244 df-fn 5245 df-f 5246 df-fv 5250 df-ov 5907 df-oprab 5908 df-mpo 5909 df-map 6684

This theorem is referenced by: bj-charfunbi 15102

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