Theorem bj-charfundcALT 14564

Description: Alternate proof of bj-charfundc 14563. It was expected to be much shorter since it uses bj-charfun 14562 for the main part of the proof and the rest is basic computations, but these turn out to be lengthy, maybe because of the limited library of available lemmas. (Contributed by BJ, 15-Aug-2024.) (Proof modification is discouraged.) (New usage is discouraged.)

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-charfundcALT	|- ( 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   x, A   x, F

Proof of Theorem bj-charfundcALT

Step	Hyp	Ref	Expression
1		bj-charfundc.1	. . 3 |- ( ph -> F = ( x e. X |-> if ( x e. A , 1o , (/) ) ) )
2	1	bj-charfun 14562	. 2 |- ( ph -> ( ( F : X --> ~P 1o /\ ( F |` ( ( X i^i A ) u. ( X \ A ) ) ) : ( ( X i^i A ) u. ( X \ A ) ) --> 2o ) /\ ( A. x e. ( X i^i A ) ( F ` x ) = 1o /\ A. x e. ( X \ A ) ( F ` x ) = (/) ) ) )
3		difin 3373	. . . . . . . . . . . 12 |- ( X \ ( X i^i A ) ) = ( X \ A )
4	3	eqcomi 2181	. . . . . . . . . . 11 |- ( X \ A ) = ( X \ ( X i^i A ) )
5	4	a1i 9	. . . . . . . . . 10 |- ( ph -> ( X \ A ) = ( X \ ( X i^i A ) ) )
6	5	uneq2d 3290	. . . . . . . . 9 |- ( ph -> ( ( X i^i A ) u. ( X \ A ) ) = ( ( X i^i A ) u. ( X \ ( X i^i A ) ) ) )
7		inss1 3356	. . . . . . . . . . 11 |- ( X i^i A ) C_ X
8	7	a1i 9	. . . . . . . . . 10 |- ( ph -> ( X i^i A ) C_ X )
9		bj-charfundc.dc	. . . . . . . . . . 11 |- ( ph -> A. x e. X DECID x e. A )
10		elin 3319	. . . . . . . . . . . . . 14 |- ( x e. ( X i^i A ) <-> ( x e. X /\ x e. A ) )
11	10	baibr 920	. . . . . . . . . . . . 13 |- ( x e. X -> ( x e. A <-> x e. ( X i^i A ) ) )
12	11	dcbid 838	. . . . . . . . . . . 12 |- ( x e. X -> (DECID x e. A <-> DECID x e. ( X i^i A ) ) )
13	12	ralbiia 2491	. . . . . . . . . . 11 |- ( A. x e. X DECID x e. A <-> A. x e. X DECID x e. ( X i^i A ) )
14	9, 13	sylib 122	. . . . . . . . . 10 |- ( ph -> A. x e. X DECID x e. ( X i^i A ) )
15		undifdcss 6922	. . . . . . . . . 10 |- ( X = ( ( X i^i A ) u. ( X \ ( X i^i A ) ) ) <-> ( ( X i^i A ) C_ X /\ A. x e. X DECID x e. ( X i^i A ) ) )
16	8, 14, 15	sylanbrc 417	. . . . . . . . 9 |- ( ph -> X = ( ( X i^i A ) u. ( X \ ( X i^i A ) ) ) )
17	6, 16	eqtr4d 2213	. . . . . . . 8 |- ( ph -> ( ( X i^i A ) u. ( X \ A ) ) = X )
18	17	reseq2d 4908	. . . . . . 7 |- ( ph -> ( F |` ( ( X i^i A ) u. ( X \ A ) ) ) = ( F |` X ) )
19		ssidd 3177	. . . . . . . . 9 |- ( ph -> X C_ X )
20	19	resmptd 4959	. . . . . . . 8 |- ( ph -> ( ( x e. X |-> if ( x e. A , 1o , (/) ) ) |` X ) = ( x e. X |-> if ( x e. A , 1o , (/) ) ) )
21	1	reseq1d 4907	. . . . . . . 8 |- ( ph -> ( F |` X ) = ( ( x e. X |-> if ( x e. A , 1o , (/) ) ) |` X ) )
22	20, 21, 1	3eqtr4d 2220	. . . . . . 7 |- ( ph -> ( F |` X ) = F )
23	18, 22	eqtrd 2210	. . . . . 6 |- ( ph -> ( F |` ( ( X i^i A ) u. ( X \ A ) ) ) = F )
24	23, 17	feq12d 5356	. . . . 5 |- ( ph -> ( ( F |` ( ( X i^i A ) u. ( X \ A ) ) ) : ( ( X i^i A ) u. ( X \ A ) ) --> 2o <-> F : X --> 2o ) )
25	24	biimpd 144	. . . 4 |- ( ph -> ( ( F |` ( ( X i^i A ) u. ( X \ A ) ) ) : ( ( X i^i A ) u. ( X \ A ) ) --> 2o -> F : X --> 2o ) )
26	25	adantld 278	. . 3 |- ( ph -> ( ( F : X --> ~P 1o /\ ( F |` ( ( X i^i A ) u. ( X \ A ) ) ) : ( ( X i^i A ) u. ( X \ A ) ) --> 2o ) -> F : X --> 2o ) )
27	26	anim1d 336	. 2 |- ( ph -> ( ( ( F : X --> ~P 1o /\ ( F |` ( ( X i^i A ) u. ( X \ A ) ) ) : ( ( X i^i A ) u. ( X \ A ) ) --> 2o ) /\ ( A. x e. ( X i^i A ) ( F ` x ) = 1o /\ A. x e. ( X \ A ) ( F ` x ) = (/) ) ) -> ( F : X --> 2o /\ ( A. x e. ( X i^i A ) ( F ` x ) = 1o /\ A. x e. ( X \ A ) ( F ` x ) = (/) ) ) ) )
28	2, 27	mpd 13	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   u. cun 3128   i^i cin 3129   C_ wss 3130   (/)c0 3423   ifcif 3535   ~Pcpw 3576   |-> cmpt 4065   |` cres 4629   -->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-fal 1359  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: (None)