Theorem bj-charfundcALT 16579

Description: Alternate proof of bj-charfundc 16578. It was expected to be much shorter since it uses bj-charfun 16577 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	⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝑋 ↦ if(𝑥 ∈ 𝐴, 1o, ∅)))
bj-charfundc.dc	⊢ (𝜑 → ∀𝑥 ∈ 𝑋 DECID 𝑥 ∈ 𝐴)

Assertion

Ref	Expression
bj-charfundcALT	⊢ (𝜑 → (𝐹:𝑋⟶2o ∧ (∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝐹‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝐹‘𝑥) = ∅)))

Distinct variable groups:   𝜑,𝑥   𝑥,𝑋   𝑥,𝐴   𝑥,𝐹

Proof of Theorem bj-charfundcALT

Step	Hyp	Ref	Expression
1		bj-charfundc.1	. . 3 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝑋 ↦ if(𝑥 ∈ 𝐴, 1o, ∅)))
2	1	bj-charfun 16577	. 2 ⊢ (𝜑 → ((𝐹:𝑋⟶𝒫 1o ∧ (𝐹 ↾ ((𝑋 ∩ 𝐴) ∪ (𝑋 ∖ 𝐴))):((𝑋 ∩ 𝐴) ∪ (𝑋 ∖ 𝐴))⟶2o) ∧ (∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝐹‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝐹‘𝑥) = ∅)))
3		difin 3458	. . . . . . . . . . . 12 ⊢ (𝑋 ∖ (𝑋 ∩ 𝐴)) = (𝑋 ∖ 𝐴)
4	3	eqcomi 2236	. . . . . . . . . . 11 ⊢ (𝑋 ∖ 𝐴) = (𝑋 ∖ (𝑋 ∩ 𝐴))
5	4	a1i 9	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 ∖ 𝐴) = (𝑋 ∖ (𝑋 ∩ 𝐴)))
6	5	uneq2d 3373	. . . . . . . . 9 ⊢ (𝜑 → ((𝑋 ∩ 𝐴) ∪ (𝑋 ∖ 𝐴)) = ((𝑋 ∩ 𝐴) ∪ (𝑋 ∖ (𝑋 ∩ 𝐴))))
7		inss1 3441	. . . . . . . . . . 11 ⊢ (𝑋 ∩ 𝐴) ⊆ 𝑋
8	7	a1i 9	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 ∩ 𝐴) ⊆ 𝑋)
9		bj-charfundc.dc	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 DECID 𝑥 ∈ 𝐴)
10		elin 3402	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (𝑋 ∩ 𝐴) ↔ (𝑥 ∈ 𝑋 ∧ 𝑥 ∈ 𝐴))
11	10	baibr 928	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑋 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ (𝑋 ∩ 𝐴)))
12	11	dcbid 846	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑋 → (DECID 𝑥 ∈ 𝐴 ↔ DECID 𝑥 ∈ (𝑋 ∩ 𝐴)))
13	12	ralbiia 2556	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ 𝑋 DECID 𝑥 ∈ 𝐴 ↔ ∀𝑥 ∈ 𝑋 DECID 𝑥 ∈ (𝑋 ∩ 𝐴))
14	9, 13	sylib 122	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 DECID 𝑥 ∈ (𝑋 ∩ 𝐴))
15		undifdcss 7183	. . . . . . . . . 10 ⊢ (𝑋 = ((𝑋 ∩ 𝐴) ∪ (𝑋 ∖ (𝑋 ∩ 𝐴))) ↔ ((𝑋 ∩ 𝐴) ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝑋 DECID 𝑥 ∈ (𝑋 ∩ 𝐴)))
16	8, 14, 15	sylanbrc 417	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 = ((𝑋 ∩ 𝐴) ∪ (𝑋 ∖ (𝑋 ∩ 𝐴))))
17	6, 16	eqtr4d 2268	. . . . . . . 8 ⊢ (𝜑 → ((𝑋 ∩ 𝐴) ∪ (𝑋 ∖ 𝐴)) = 𝑋)
18	17	reseq2d 5038	. . . . . . 7 ⊢ (𝜑 → (𝐹 ↾ ((𝑋 ∩ 𝐴) ∪ (𝑋 ∖ 𝐴))) = (𝐹 ↾ 𝑋))
19		ssidd 3259	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ⊆ 𝑋)
20	19	resmptd 5089	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ if(𝑥 ∈ 𝐴, 1o, ∅)) ↾ 𝑋) = (𝑥 ∈ 𝑋 ↦ if(𝑥 ∈ 𝐴, 1o, ∅)))
21	1	reseq1d 5037	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ↾ 𝑋) = ((𝑥 ∈ 𝑋 ↦ if(𝑥 ∈ 𝐴, 1o, ∅)) ↾ 𝑋))
22	20, 21, 1	3eqtr4d 2275	. . . . . . 7 ⊢ (𝜑 → (𝐹 ↾ 𝑋) = 𝐹)
23	18, 22	eqtrd 2265	. . . . . 6 ⊢ (𝜑 → (𝐹 ↾ ((𝑋 ∩ 𝐴) ∪ (𝑋 ∖ 𝐴))) = 𝐹)
24	23, 17	feq12d 5498	. . . . 5 ⊢ (𝜑 → ((𝐹 ↾ ((𝑋 ∩ 𝐴) ∪ (𝑋 ∖ 𝐴))):((𝑋 ∩ 𝐴) ∪ (𝑋 ∖ 𝐴))⟶2o ↔ 𝐹:𝑋⟶2o))
25	24	biimpd 144	. . . 4 ⊢ (𝜑 → ((𝐹 ↾ ((𝑋 ∩ 𝐴) ∪ (𝑋 ∖ 𝐴))):((𝑋 ∩ 𝐴) ∪ (𝑋 ∖ 𝐴))⟶2o → 𝐹:𝑋⟶2o))
26	25	adantld 278	. . 3 ⊢ (𝜑 → ((𝐹:𝑋⟶𝒫 1o ∧ (𝐹 ↾ ((𝑋 ∩ 𝐴) ∪ (𝑋 ∖ 𝐴))):((𝑋 ∩ 𝐴) ∪ (𝑋 ∖ 𝐴))⟶2o) → 𝐹:𝑋⟶2o))
27	26	anim1d 336	. 2 ⊢ (𝜑 → (((𝐹:𝑋⟶𝒫 1o ∧ (𝐹 ↾ ((𝑋 ∩ 𝐴) ∪ (𝑋 ∖ 𝐴))):((𝑋 ∩ 𝐴) ∪ (𝑋 ∖ 𝐴))⟶2o) ∧ (∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝐹‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝐹‘𝑥) = ∅)) → (𝐹:𝑋⟶2o ∧ (∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝐹‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝐹‘𝑥) = ∅))))
28	2, 27	mpd 13	1 ⊢ (𝜑 → (𝐹:𝑋⟶2o ∧ (∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝐹‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝐹‘𝑥) = ∅)))

Syntax hints:   → wi 4   ∧ wa 104  DECID wdc 842   = wceq 1398   ∈ wcel 2203  ∀wral 2520   ∖ cdif 3208   ∪ cun 3209   ∩ cin 3210   ⊆ wss 3211  ∅c0 3508  ifcif 3620  𝒫 cpw 3669   ↦ cmpt 4171   ↾ cres 4751  ⟶wf 5348  ‘cfv 5352  1_oc1o 6640  2_oc2o 6641

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-suc 4492  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-fv 5360  df-1o 6647  df-2o 6648

This theorem is referenced by: (None)