Theorem bj-charfundc 14563

Description: Properties of the characteristic function on the class X of the class A, provided membership in A is decidable in X. (Contributed by BJ, 6-Aug-2024.)

Hypotheses

Ref	Expression
bj-charfundc.1	|- ( ph -> F = ( x e. X |-> if ( x e. A , 1o , (/) ) ) )
bj-charfundc.dc	|- ( ph -> A. x e. X DECID x e. A )

Assertion

Ref	Expression
bj-charfundc	|- ( ph -> ( F : X --> 2o /\ ( A. x e. ( X i^i A ) ( F ` x ) = 1o /\ A. x e. ( X \ A ) ( F ` x ) = (/) ) ) )

Distinct variable groups:   ph, x   x, X

Allowed substitution hints:   A( x)   F( x)

Proof of Theorem bj-charfundc

Step	Hyp	Ref	Expression
1		bj-charfundc.1	. . 3 |- ( ph -> F = ( x e. X |-> if ( x e. A , 1o , (/) ) ) )
2		1lt2o 6443	. . . . 5 |- 1o e. 2o
3	2	a1i 9	. . . 4 |- ( ( ph /\ x e. X ) -> 1o e. 2o )
4		0lt2o 6442	. . . . 5 |- (/) e. 2o
5	4	a1i 9	. . . 4 |- ( ( ph /\ x e. X ) -> (/) e. 2o )
6		bj-charfundc.dc	. . . . 5 |- ( ph -> A. x e. X DECID x e. A )
7	6	r19.21bi 2565	. . . 4 |- ( ( ph /\ x e. X ) -> DECID x e. A )
8	3, 5, 7	ifcldcd 3571	. . 3 |- ( ( ph /\ x e. X ) -> if ( x e. A , 1o , (/) ) e. 2o )
9	1, 8	fmpt3d 5673	. 2 |- ( ph -> F : X --> 2o )
10		inss1 3356	. . . . . . . 8 |- ( X i^i A ) C_ X
11	10	a1i 9	. . . . . . 7 |- ( ph -> ( X i^i A ) C_ X )
12	11	sseld 3155	. . . . . 6 |- ( ph -> ( x e. ( X i^i A ) -> x e. X ) )
13	12	imdistani 445	. . . . 5 |- ( ( ph /\ x e. ( X i^i A ) ) -> ( ph /\ x e. X ) )
14	1, 8	fvmpt2d 5603	. . . . 5 |- ( ( ph /\ x e. X ) -> ( F ` x ) = if ( x e. A , 1o , (/) ) )
15	13, 14	syl 14	. . . 4 |- ( ( ph /\ x e. ( X i^i A ) ) -> ( F ` x ) = if ( x e. A , 1o , (/) ) )
16		simpr 110	. . . . . 6 |- ( ( ph /\ x e. ( X i^i A ) ) -> x e. ( X i^i A ) )
17	16	elin2d 3326	. . . . 5 |- ( ( ph /\ x e. ( X i^i A ) ) -> x e. A )
18	17	iftrued 3542	. . . 4 |- ( ( ph /\ x e. ( X i^i A ) ) -> if ( x e. A , 1o , (/) ) = 1o )
19	15, 18	eqtrd 2210	. . 3 |- ( ( ph /\ x e. ( X i^i A ) ) -> ( F ` x ) = 1o )
20	19	ralrimiva 2550	. 2 |- ( ph -> A. x e. ( X i^i A ) ( F ` x ) = 1o )
21		difssd 3263	. . . . . . 7 |- ( ph -> ( X \ A ) C_ X )
22	21	sseld 3155	. . . . . 6 |- ( ph -> ( x e. ( X \ A ) -> x e. X ) )
23	22	imdistani 445	. . . . 5 |- ( ( ph /\ x e. ( X \ A ) ) -> ( ph /\ x e. X ) )
24	23, 14	syl 14	. . . 4 |- ( ( ph /\ x e. ( X \ A ) ) -> ( F ` x ) = if ( x e. A , 1o , (/) ) )
25		simpr 110	. . . . . 6 |- ( ( ph /\ x e. ( X \ A ) ) -> x e. ( X \ A ) )
26	25	eldifbd 3142	. . . . 5 |- ( ( ph /\ x e. ( X \ A ) ) -> -. x e. A )
27	26	iffalsed 3545	. . . 4 |- ( ( ph /\ x e. ( X \ A ) ) -> if ( x e. A , 1o , (/) ) = (/) )
28	24, 27	eqtrd 2210	. . 3 |- ( ( ph /\ x e. ( X \ A ) ) -> ( F ` x ) = (/) )
29	28	ralrimiva 2550	. 2 |- ( ph -> A. x e. ( X \ A ) ( F ` x ) = (/) )
30	9, 20, 29	jca32 310	1 |- ( ph -> ( F : X --> 2o /\ ( A. x e. ( X i^i A ) ( F ` x ) = 1o /\ A. x e. ( X \ A ) ( F ` x ) = (/) ) ) )

Syntax hints:   -> wi 4   /\ wa 104  DECID wdc 834   = wceq 1353   e. wcel 2148   A.wral 2455   \ cdif 3127   i^i cin 3129   C_ wss 3130   (/)c0 3423   ifcif 3535   |-> cmpt 4065   -->wf 5213   ` cfv 5217   1oc1o 6410   2oc2o 6411

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

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3an 980  df-tru 1356  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-ral 2460  df-rex 2461  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-iord 4367  df-on 4369  df-suc 4372  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-fv 5225  df-1o 6417  df-2o 6418

This theorem is referenced by:  bj-charfunbi  14566