Theorem bj-charfundcALT 15539

Description: Alternate proof of bj-charfundc 15538. It was expected to be much shorter since it uses bj-charfun 15537 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 15537	. 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 3401	. . . . . . . . . . . 12 |- ( X \ ( X i^i A ) ) = ( X \ A )
4	3	eqcomi 2200	. . . . . . . . . . 11 |- ( X \ A ) = ( X \ ( X i^i A ) )
5	4	a1i 9	. . . . . . . . . 10 |- ( ph -> ( X \ A ) = ( X \ ( X i^i A ) ) )
6	5	uneq2d 3318	. . . . . . . . 9 |- ( ph -> ( ( X i^i A ) u. ( X \ A ) ) = ( ( X i^i A ) u. ( X \ ( X i^i A ) ) ) )
7		inss1 3384	. . . . . . . . . . 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 3347	. . . . . . . . . . . . . 14 |- ( x e. ( X i^i A ) <-> ( x e. X /\ x e. A ) )
11	10	baibr 921	. . . . . . . . . . . . 13 |- ( x e. X -> ( x e. A <-> x e. ( X i^i A ) ) )
12	11	dcbid 839	. . . . . . . . . . . 12 |- ( x e. X -> (DECID x e. A <-> DECID x e. ( X i^i A ) ) )
13	12	ralbiia 2511	. . . . . . . . . . 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 6993	. . . . . . . . . 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 2232	. . . . . . . 8 |- ( ph -> ( ( X i^i A ) u. ( X \ A ) ) = X )
18	17	reseq2d 4947	. . . . . . 7 |- ( ph -> ( F |` ( ( X i^i A ) u. ( X \ A ) ) ) = ( F |` X ) )
19		ssidd 3205	. . . . . . . . 9 |- ( ph -> X C_ X )
20	19	resmptd 4998	. . . . . . . 8 |- ( ph -> ( ( x e. X |-> if ( x e. A , 1o , (/) ) ) |` X ) = ( x e. X |-> if ( x e. A , 1o , (/) ) ) )
21	1	reseq1d 4946	. . . . . . . 8 |- ( ph -> ( F |` X ) = ( ( x e. X |-> if ( x e. A , 1o , (/) ) ) |` X ) )
22	20, 21, 1	3eqtr4d 2239	. . . . . . 7 |- ( ph -> ( F |` X ) = F )
23	18, 22	eqtrd 2229	. . . . . 6 |- ( ph -> ( F |` ( ( X i^i A ) u. ( X \ A ) ) ) = F )
24	23, 17	feq12d 5400	. . . . 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 835   = wceq 1364   e. wcel 2167   A.wral 2475   \ cdif 3154   u. cun 3155   i^i cin 3156   C_ wss 3157   (/)c0 3451   ifcif 3562   ~Pcpw 3606   |-> cmpt 4095   |` cres 4666   -->wf 5255   ` cfv 5259   1oc1o 6476   2oc2o 6477

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-iord 4402  df-on 4404  df-suc 4407  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-fv 5267  df-1o 6483  df-2o 6484

This theorem is referenced by: (None)