Theorem bj-charfundc 16748

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 6705	. . . . 5 |- 1o e. 2o
3	2	a1i 9	. . . 4 |- ( ( ph /\ x e. X ) -> 1o e. 2o )
4		0lt2o 6704	. . . . 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 2638	. . . 4 |- ( ( ph /\ x e. X ) -> DECID x e. A )
8	3, 5, 7	ifcldcd 3675	. . 3 |- ( ( ph /\ x e. X ) -> if ( x e. A , 1o , (/) ) e. 2o )
9	1, 8	fmpt3d 5855	. 2 |- ( ph -> F : X --> 2o )
10		inss1 3451	. . . . . . . 8 |- ( X i^i A ) C_ X
11	10	a1i 9	. . . . . . 7 |- ( ph -> ( X i^i A ) C_ X )
12	11	sseld 3247	. . . . . 6 |- ( ph -> ( x e. ( X i^i A ) -> x e. X ) )
13	12	imdistani 449	. . . . 5 |- ( ( ph /\ x e. ( X i^i A ) ) -> ( ph /\ x e. X ) )
14	1, 8	fvmpt2d 5786	. . . . 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 3419	. . . . 5 |- ( ( ph /\ x e. ( X i^i A ) ) -> x e. A )
18	17	iftrued 3644	. . . 4 |- ( ( ph /\ x e. ( X i^i A ) ) -> if ( x e. A , 1o , (/) ) = 1o )
19	15, 18	eqtrd 2271	. . 3 |- ( ( ph /\ x e. ( X i^i A ) ) -> ( F ` x ) = 1o )
20	19	ralrimiva 2623	. 2 |- ( ph -> A. x e. ( X i^i A ) ( F ` x ) = 1o )
21		difssd 3356	. . . . . . 7 |- ( ph -> ( X \ A ) C_ X )
22	21	sseld 3247	. . . . . 6 |- ( ph -> ( x e. ( X \ A ) -> x e. X ) )
23	22	imdistani 449	. . . . 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 3232	. . . . 5 |- ( ( ph /\ x e. ( X \ A ) ) -> -. x e. A )
27	26	iffalsed 3647	. . . 4 |- ( ( ph /\ x e. ( X \ A ) ) -> if ( x e. A , 1o , (/) ) = (/) )
28	24, 27	eqtrd 2271	. . 3 |- ( ( ph /\ x e. ( X \ A ) ) -> ( F ` x ) = (/) )
29	28	ralrimiva 2623	. 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 846   = wceq 1402   e. wcel 2209   A.wral 2528   \ cdif 3217   i^i cin 3219   C_ wss 3220   (/)c0 3520   ifcif 3635   |-> cmpt 4187   -->wf 5368   ` cfv 5372   1oc1o 6670   2oc2o 6671

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-suc 4511  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fv 5380  df-1o 6677  df-2o 6678

This theorem is referenced by:  bj-charfunbi  16751