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Theorem bj-charfunr 13845

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 6648	. . . . . . . . . 10
3		ffvelrn 5629	. . . . . . . . . . 11 $/\$
4	3	ex 114	. . . . . . . . . 10
5	2, 4	syl 14	. . . . . . . . 9
6		0elnn 4603	. . . . . . . . . 10 $\/$
7		nn0eln0 4604	. . . . . . . . . . 11
8	7	orbi2d 785	. . . . . . . . . 10 $\/$ $\/$
9	6, 8	mpbid 146	. . . . . . . . 9 $\/$
10	5, 9	syl6 33	. . . . . . . 8 $\/$
11	10	adantr 274	. . . . . . 7 $/\$ $/\$ $\$ $\/$
12		elin 3310	. . . . . . . . . . . . . . 15 $/\$
13		rsp 2517	. . . . . . . . . . . . . . 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 2398	. . . . . . . . . 10 $/\$ $\$ $/\$
19		eldif 3130	. . . . . . . . . . . . . . 15 $\$ $/\$
20		rsp 2517	. . . . . . . . . . . . . . 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 2382	. . . . . . . . . 10 $/\$ $\$ $/\$
26	18, 25	orim12d 781	. . . . . . . . 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 2592	. . . 4 $\/$
32	31	imp 123	. . 3 $/\$ $\/$
33		df-dc 830	. . 3 DECID $\/$
34	32, 33	sylibr 133	. 2 $/\$ DECID
35	34	ralrimiva 2543	1 DECID

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 103 $\/$ wo 703 DECID wdc 829

wceq 1348

wcel 2141

wne 2340

wral 2448

wrex 2449 $\$ cdif 3118

cin 3120

c0 3414

com 4574

wf 5194

cfv 5198 (class class class)co 5853

cmap 6626

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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572

This theorem depends on definitions: df-bi 116 df-dc 830 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-id 4278 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-fv 5206 df-ov 5856 df-oprab 5857 df-mpo 5858 df-map 6628

This theorem is referenced by: bj-charfunbi 13846

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