Theorem bj-charfundc 16704

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 6688	. . . . 5 |- 1o e. 2o
3	2	a1i 9	. . . 4 |- ( ( ph /\ x e. X ) -> 1o e. 2o )
4		0lt2o 6687	. . . . 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 2632	. . . 4 |- ( ( ph /\ x e. X ) -> DECID x e. A )
8	3, 5, 7	ifcldcd 3664	. . 3 |- ( ( ph /\ x e. X ) -> if ( x e. A , 1o , (/) ) e. 2o )
9	1, 8	fmpt3d 5838	. 2 |- ( ph -> F : X --> 2o )
10		inss1 3445	. . . . . . . 8 |- ( X i^i A ) C_ X
11	10	a1i 9	. . . . . . 7 |- ( ph -> ( X i^i A ) C_ X )
12	11	sseld 3241	. . . . . 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 5769	. . . . 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 3413	. . . . 5 |- ( ( ph /\ x e. ( X i^i A ) ) -> x e. A )
18	17	iftrued 3633	. . . 4 |- ( ( ph /\ x e. ( X i^i A ) ) -> if ( x e. A , 1o , (/) ) = 1o )
19	15, 18	eqtrd 2267	. . 3 |- ( ( ph /\ x e. ( X i^i A ) ) -> ( F ` x ) = 1o )
20	19	ralrimiva 2617	. 2 |- ( ph -> A. x e. ( X i^i A ) ( F ` x ) = 1o )
21		difssd 3350	. . . . . . 7 |- ( ph -> ( X \ A ) C_ X )
22	21	sseld 3241	. . . . . 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 3226	. . . . 5 |- ( ( ph /\ x e. ( X \ A ) ) -> -. x e. A )
27	26	iffalsed 3636	. . . 4 |- ( ( ph /\ x e. ( X \ A ) ) -> if ( x e. A , 1o , (/) ) = (/) )
28	24, 27	eqtrd 2267	. . 3 |- ( ( ph /\ x e. ( X \ A ) ) -> ( F ` x ) = (/) )
29	28	ralrimiva 2617	. 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 842   = wceq 1398   e. wcel 2205   A.wral 2522   \ cdif 3211   i^i cin 3213   C_ wss 3214   (/)c0 3512   ifcif 3624   |-> cmpt 4176   -->wf 5353   ` cfv 5357   1oc1o 6653   2oc2o 6654

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-suc 4497  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fv 5365  df-1o 6660  df-2o 6661

This theorem is referenced by:  bj-charfunbi  16707