Theorem bj-charfundcALT 15944

Description: Alternate proof of bj-charfundc 15943. It was expected to be much shorter since it uses bj-charfun 15942 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 15942	. 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 3418	. . . . . . . . . . . 12 |- ( X \ ( X i^i A ) ) = ( X \ A )
4	3	eqcomi 2211	. . . . . . . . . . 11 |- ( X \ A ) = ( X \ ( X i^i A ) )
5	4	a1i 9	. . . . . . . . . 10 |- ( ph -> ( X \ A ) = ( X \ ( X i^i A ) ) )
6	5	uneq2d 3335	. . . . . . . . 9 |- ( ph -> ( ( X i^i A ) u. ( X \ A ) ) = ( ( X i^i A ) u. ( X \ ( X i^i A ) ) ) )
7		inss1 3401	. . . . . . . . . . 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 3364	. . . . . . . . . . . . . 14 |- ( x e. ( X i^i A ) <-> ( x e. X /\ x e. A ) )
11	10	baibr 922	. . . . . . . . . . . . 13 |- ( x e. X -> ( x e. A <-> x e. ( X i^i A ) ) )
12	11	dcbid 840	. . . . . . . . . . . 12 |- ( x e. X -> (DECID x e. A <-> DECID x e. ( X i^i A ) ) )
13	12	ralbiia 2522	. . . . . . . . . . 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 7046	. . . . . . . . . 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 2243	. . . . . . . 8 |- ( ph -> ( ( X i^i A ) u. ( X \ A ) ) = X )
18	17	reseq2d 4978	. . . . . . 7 |- ( ph -> ( F |` ( ( X i^i A ) u. ( X \ A ) ) ) = ( F |` X ) )
19		ssidd 3222	. . . . . . . . 9 |- ( ph -> X C_ X )
20	19	resmptd 5029	. . . . . . . 8 |- ( ph -> ( ( x e. X |-> if ( x e. A , 1o , (/) ) ) |` X ) = ( x e. X |-> if ( x e. A , 1o , (/) ) ) )
21	1	reseq1d 4977	. . . . . . . 8 |- ( ph -> ( F |` X ) = ( ( x e. X |-> if ( x e. A , 1o , (/) ) ) |` X ) )
22	20, 21, 1	3eqtr4d 2250	. . . . . . 7 |- ( ph -> ( F |` X ) = F )
23	18, 22	eqtrd 2240	. . . . . 6 |- ( ph -> ( F |` ( ( X i^i A ) u. ( X \ A ) ) ) = F )
24	23, 17	feq12d 5435	. . . . 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 836   = wceq 1373   e. wcel 2178   A.wral 2486   \ cdif 3171   u. cun 3172   i^i cin 3173   C_ wss 3174   (/)c0 3468   ifcif 3579   ~Pcpw 3626   |-> cmpt 4121   |` cres 4695   -->wf 5286   ` cfv 5290   1oc1o 6518   2oc2o 6519

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-iord 4431  df-on 4433  df-suc 4436  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-fv 5298  df-1o 6525  df-2o 6526

This theorem is referenced by: (None)