Theorem bj-charfundcALT 16596

Description: Alternate proof of bj-charfundc 16595. It was expected to be much shorter since it uses bj-charfun 16594 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 16594	. 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 3460	. . . . . . . . . . . 12 |- ( X \ ( X i^i A ) ) = ( X \ A )
4	3	eqcomi 2238	. . . . . . . . . . 11 |- ( X \ A ) = ( X \ ( X i^i A ) )
5	4	a1i 9	. . . . . . . . . 10 |- ( ph -> ( X \ A ) = ( X \ ( X i^i A ) ) )
6	5	uneq2d 3375	. . . . . . . . 9 |- ( ph -> ( ( X i^i A ) u. ( X \ A ) ) = ( ( X i^i A ) u. ( X \ ( X i^i A ) ) ) )
7		inss1 3443	. . . . . . . . . . 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 3404	. . . . . . . . . . . . . 14 |- ( x e. ( X i^i A ) <-> ( x e. X /\ x e. A ) )
11	10	baibr 928	. . . . . . . . . . . . 13 |- ( x e. X -> ( x e. A <-> x e. ( X i^i A ) ) )
12	11	dcbid 846	. . . . . . . . . . . 12 |- ( x e. X -> (DECID x e. A <-> DECID x e. ( X i^i A ) ) )
13	12	ralbiia 2558	. . . . . . . . . . 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 7185	. . . . . . . . . 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 2270	. . . . . . . 8 |- ( ph -> ( ( X i^i A ) u. ( X \ A ) ) = X )
18	17	reseq2d 5040	. . . . . . 7 |- ( ph -> ( F |` ( ( X i^i A ) u. ( X \ A ) ) ) = ( F |` X ) )
19		ssidd 3261	. . . . . . . . 9 |- ( ph -> X C_ X )
20	19	resmptd 5091	. . . . . . . 8 |- ( ph -> ( ( x e. X |-> if ( x e. A , 1o , (/) ) ) |` X ) = ( x e. X |-> if ( x e. A , 1o , (/) ) ) )
21	1	reseq1d 5039	. . . . . . . 8 |- ( ph -> ( F |` X ) = ( ( x e. X |-> if ( x e. A , 1o , (/) ) ) |` X ) )
22	20, 21, 1	3eqtr4d 2277	. . . . . . 7 |- ( ph -> ( F |` X ) = F )
23	18, 22	eqtrd 2267	. . . . . 6 |- ( ph -> ( F |` ( ( X i^i A ) u. ( X \ A ) ) ) = F )
24	23, 17	feq12d 5500	. . . . 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 842   = wceq 1398   e. wcel 2205   A.wral 2522   \ cdif 3210   u. cun 3211   i^i cin 3212   C_ wss 3213   (/)c0 3510   ifcif 3622   ~Pcpw 3671   |-> cmpt 4173   |` cres 4753   -->wf 5350   ` cfv 5354   1oc1o 6642   2oc2o 6643

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 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3an 1007  df-tru 1401  df-fal 1404  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 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-iord 4489  df-on 4491  df-suc 4494  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-fv 5362  df-1o 6649  df-2o 6650

This theorem is referenced by: (None)