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Theorem bj-charfunr 15746

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 6757	. . . . . . . . . 10
3		ffvelcdm 5713	. . . . . . . . . . 11 $/\$
4	3	ex 115	. . . . . . . . . 10
5	2, 4	syl 14	. . . . . . . . 9
6		0elnn 4667	. . . . . . . . . 10 $\/$
7		nn0eln0 4668	. . . . . . . . . . 11
8	7	orbi2d 792	. . . . . . . . . 10 $\/$ $\/$
9	6, 8	mpbid 147	. . . . . . . . 9 $\/$
10	5, 9	syl6 33	. . . . . . . 8 $\/$
11	10	adantr 276	. . . . . . 7 $/\$ $/\$ $\$ $\/$
12		elin 3356	. . . . . . . . . . . . . . 15 $/\$
13		rsp 2553	. . . . . . . . . . . . . . 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 2434	. . . . . . . . . 10 $/\$ $\$ $/\$
19		eldif 3175	. . . . . . . . . . . . . . 15 $\$ $/\$
20		rsp 2553	. . . . . . . . . . . . . . 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 2418	. . . . . . . . . 10 $/\$ $\$ $/\$
26	18, 25	orim12d 788	. . . . . . . . 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 2628	. . . 4 $\/$
32	31	imp 124	. . 3 $/\$ $\/$
33		df-dc 837	. . 3 DECID $\/$
34	32, 33	sylibr 134	. 2 $/\$ DECID
35	34	ralrimiva 2579	1 DECID

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104 $\/$ wo 710 DECID wdc 836

wceq 1373

wcel 2176

wne 2376

wral 2484

wrex 2485 $\$ cdif 3163

cin 3165

c0 3460

com 4638

wf 5267

cfv 5271 (class class class)co 5944

cmap 6735

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 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-sep 4162 ax-nul 4170 ax-pow 4218 ax-pr 4253 ax-un 4480 ax-setind 4585 ax-iinf 4636

This theorem depends on definitions: df-bi 117 df-dc 837 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ne 2377 df-ral 2489 df-rex 2490 df-v 2774 df-sbc 2999 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-nul 3461 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-int 3886 df-br 4045 df-opab 4106 df-id 4340 df-suc 4418 df-iom 4639 df-xp 4681 df-rel 4682 df-cnv 4683 df-co 4684 df-dm 4685 df-rn 4686 df-iota 5232 df-fun 5273 df-fn 5274 df-f 5275 df-fv 5279 df-ov 5947 df-oprab 5948 df-mpo 5949 df-map 6737

This theorem is referenced by: bj-charfunbi 15747

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