Theorem bj-charfundcALT 13844

Description: Alternate proof of bj-charfundc 13843. It was expected to be much shorter since it uses bj-charfun 13842 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 13842	. 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 3364	. . . . . . . . . . . 12 |- ( X \ ( X i^i A ) ) = ( X \ A )
4	3	eqcomi 2174	. . . . . . . . . . 11 |- ( X \ A ) = ( X \ ( X i^i A ) )
5	4	a1i 9	. . . . . . . . . 10 |- ( ph -> ( X \ A ) = ( X \ ( X i^i A ) ) )
6	5	uneq2d 3281	. . . . . . . . 9 |- ( ph -> ( ( X i^i A ) u. ( X \ A ) ) = ( ( X i^i A ) u. ( X \ ( X i^i A ) ) ) )
7		inss1 3347	. . . . . . . . . . 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 3310	. . . . . . . . . . . . . 14 |- ( x e. ( X i^i A ) <-> ( x e. X /\ x e. A ) )
11	10	baibr 915	. . . . . . . . . . . . 13 |- ( x e. X -> ( x e. A <-> x e. ( X i^i A ) ) )
12	11	dcbid 833	. . . . . . . . . . . 12 |- ( x e. X -> (DECID x e. A <-> DECID x e. ( X i^i A ) ) )
13	12	ralbiia 2484	. . . . . . . . . . 11 |- ( A. x e. X DECID x e. A <-> A. x e. X DECID x e. ( X i^i A ) )
14	9, 13	sylib 121	. . . . . . . . . 10 |- ( ph -> A. x e. X DECID x e. ( X i^i A ) )
15		undifdcss 6900	. . . . . . . . . 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 415	. . . . . . . . 9 |- ( ph -> X = ( ( X i^i A ) u. ( X \ ( X i^i A ) ) ) )
17	6, 16	eqtr4d 2206	. . . . . . . 8 |- ( ph -> ( ( X i^i A ) u. ( X \ A ) ) = X )
18	17	reseq2d 4891	. . . . . . 7 |- ( ph -> ( F |` ( ( X i^i A ) u. ( X \ A ) ) ) = ( F |` X ) )
19		ssidd 3168	. . . . . . . . 9 |- ( ph -> X C_ X )
20	19	resmptd 4942	. . . . . . . 8 |- ( ph -> ( ( x e. X |-> if ( x e. A , 1o , (/) ) ) |` X ) = ( x e. X |-> if ( x e. A , 1o , (/) ) ) )
21	1	reseq1d 4890	. . . . . . . 8 |- ( ph -> ( F |` X ) = ( ( x e. X |-> if ( x e. A , 1o , (/) ) ) |` X ) )
22	20, 21, 1	3eqtr4d 2213	. . . . . . 7 |- ( ph -> ( F |` X ) = F )
23	18, 22	eqtrd 2203	. . . . . 6 |- ( ph -> ( F |` ( ( X i^i A ) u. ( X \ A ) ) ) = F )
24	23, 17	feq12d 5337	. . . . 5 |- ( ph -> ( ( F |` ( ( X i^i A ) u. ( X \ A ) ) ) : ( ( X i^i A ) u. ( X \ A ) ) --> 2o <-> F : X --> 2o ) )
25	24	biimpd 143	. . . 4 |- ( ph -> ( ( F |` ( ( X i^i A ) u. ( X \ A ) ) ) : ( ( X i^i A ) u. ( X \ A ) ) --> 2o -> F : X --> 2o ) )
26	25	adantld 276	. . 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 334	. 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 103  DECID wdc 829   = wceq 1348   e. wcel 2141   A.wral 2448   \ cdif 3118   u. cun 3119   i^i cin 3120   C_ wss 3121   (/)c0 3414   ifcif 3526   ~Pcpw 3566   |-> cmpt 4050   |` cres 4613   -->wf 5194   ` cfv 5198   1oc1o 6388   2oc2o 6389

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-1o 6395  df-2o 6396

This theorem is referenced by: (None)