Theorem bj-charfundc 13343

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 6383	. . . . 5 |- 1o e. 2o
3	2	a1i 9	. . . 4 |- ( ( ph /\ x e. X ) -> 1o e. 2o )
4		0lt2o 6382	. . . . 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 2545	. . . 4 |- ( ( ph /\ x e. X ) -> DECID x e. A )
8	3, 5, 7	ifcldcd 3540	. . 3 |- ( ( ph /\ x e. X ) -> if ( x e. A , 1o , (/) ) e. 2o )
9	1, 8	fmpt3d 5620	. 2 |- ( ph -> F : X --> 2o )
10		inss1 3327	. . . . . . . 8 |- ( X i^i A ) C_ X
11	10	a1i 9	. . . . . . 7 |- ( ph -> ( X i^i A ) C_ X )
12	11	sseld 3127	. . . . . 6 |- ( ph -> ( x e. ( X i^i A ) -> x e. X ) )
13	12	imdistani 442	. . . . 5 |- ( ( ph /\ x e. ( X i^i A ) ) -> ( ph /\ x e. X ) )
14	1, 8	fvmpt2d 5551	. . . . 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 109	. . . . . 6 |- ( ( ph /\ x e. ( X i^i A ) ) -> x e. ( X i^i A ) )
17	16	elin2d 3297	. . . . 5 |- ( ( ph /\ x e. ( X i^i A ) ) -> x e. A )
18	17	iftrued 3512	. . . 4 |- ( ( ph /\ x e. ( X i^i A ) ) -> if ( x e. A , 1o , (/) ) = 1o )
19	15, 18	eqtrd 2190	. . 3 |- ( ( ph /\ x e. ( X i^i A ) ) -> ( F ` x ) = 1o )
20	19	ralrimiva 2530	. 2 |- ( ph -> A. x e. ( X i^i A ) ( F ` x ) = 1o )
21		difssd 3234	. . . . . . 7 |- ( ph -> ( X \ A ) C_ X )
22	21	sseld 3127	. . . . . 6 |- ( ph -> ( x e. ( X \ A ) -> x e. X ) )
23	22	imdistani 442	. . . . 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 109	. . . . . 6 |- ( ( ph /\ x e. ( X \ A ) ) -> x e. ( X \ A ) )
26	25	eldifbd 3114	. . . . 5 |- ( ( ph /\ x e. ( X \ A ) ) -> -. x e. A )
27	26	iffalsed 3515	. . . 4 |- ( ( ph /\ x e. ( X \ A ) ) -> if ( x e. A , 1o , (/) ) = (/) )
28	24, 27	eqtrd 2190	. . 3 |- ( ( ph /\ x e. ( X \ A ) ) -> ( F ` x ) = (/) )
29	28	ralrimiva 2530	. 2 |- ( ph -> A. x e. ( X \ A ) ( F ` x ) = (/) )
30	9, 20, 29	jca32 308	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 103  DECID wdc 820   = wceq 1335   e. wcel 2128   A.wral 2435   \ cdif 3099   i^i cin 3101   C_ wss 3102   (/)c0 3394   ifcif 3505   |-> cmpt 4025   -->wf 5163   ` cfv 5167   1oc1o 6350   2oc2o 6351

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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-nul 4090  ax-pow 4134  ax-pr 4168  ax-un 4392

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4252  df-iord 4325  df-on 4327  df-suc 4330  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-rn 4594  df-res 4595  df-ima 4596  df-iota 5132  df-fun 5169  df-fn 5170  df-f 5171  df-fv 5175  df-1o 6357  df-2o 6358

This theorem is referenced by:  bj-charfunbi  13346