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Theorem bj-charfunr 14565

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 6670	. . . . . . . . . 10
3		ffvelcdm 5650	. . . . . . . . . . 11 $/\$
4	3	ex 115	. . . . . . . . . 10
5	2, 4	syl 14	. . . . . . . . 9
6		0elnn 4619	. . . . . . . . . 10 $\/$
7		nn0eln0 4620	. . . . . . . . . . 11
8	7	orbi2d 790	. . . . . . . . . 10 $\/$ $\/$
9	6, 8	mpbid 147	. . . . . . . . 9 $\/$
10	5, 9	syl6 33	. . . . . . . 8 $\/$
11	10	adantr 276	. . . . . . 7 $/\$ $/\$ $\$ $\/$
12		elin 3319	. . . . . . . . . . . . . . 15 $/\$
13		rsp 2524	. . . . . . . . . . . . . . 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 2405	. . . . . . . . . 10 $/\$ $\$ $/\$
19		eldif 3139	. . . . . . . . . . . . . . 15 $\$ $/\$
20		rsp 2524	. . . . . . . . . . . . . . 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 2389	. . . . . . . . . 10 $/\$ $\$ $/\$
26	18, 25	orim12d 786	. . . . . . . . 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 2599	. . . 4 $\/$
32	31	imp 124	. . 3 $/\$ $\/$
33		df-dc 835	. . 3 DECID $\/$
34	32, 33	sylibr 134	. 2 $/\$ DECID
35	34	ralrimiva 2550	1 DECID

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104 $\/$ wo 708 DECID wdc 834

wceq 1353

wcel 2148

wne 2347

wral 2455

wrex 2456 $\$ cdif 3127

cin 3129

c0 3423

com 4590

wf 5213

cfv 5217 (class class class)co 5875

cmap 6648

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4122 ax-nul 4130 ax-pow 4175 ax-pr 4210 ax-un 4434 ax-setind 4537 ax-iinf 4588

This theorem depends on definitions: df-bi 117 df-dc 835 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-v 2740 df-sbc 2964 df-dif 3132 df-un 3134 df-in 3136 df-ss 3143 df-nul 3424 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-int 3846 df-br 4005 df-opab 4066 df-id 4294 df-suc 4372 df-iom 4591 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-rn 4638 df-iota 5179 df-fun 5219 df-fn 5220 df-f 5221 df-fv 5225 df-ov 5878 df-oprab 5879 df-mpo 5880 df-map 6650

This theorem is referenced by: bj-charfunbi 14566

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