Theorem bj-charfundcALT 16404

Description: Alternate proof of bj-charfundc 16403. It was expected to be much shorter since it uses bj-charfun 16402 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 16402	. 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 3444	. . . . . . . . . . . 12 ⊢ (𝑋 ∖ (𝑋 ∩ 𝐴)) = (𝑋 ∖ 𝐴)
4	3	eqcomi 2235	. . . . . . . . . . 11 ⊢ (𝑋 ∖ 𝐴) = (𝑋 ∖ (𝑋 ∩ 𝐴))
5	4	a1i 9	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 ∖ 𝐴) = (𝑋 ∖ (𝑋 ∩ 𝐴)))
6	5	uneq2d 3361	. . . . . . . . 9 ⊢ (𝜑 → ((𝑋 ∩ 𝐴) ∪ (𝑋 ∖ 𝐴)) = ((𝑋 ∩ 𝐴) ∪ (𝑋 ∖ (𝑋 ∩ 𝐴))))
7		inss1 3427	. . . . . . . . . . 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 3390	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (𝑋 ∩ 𝐴) ↔ (𝑥 ∈ 𝑋 ∧ 𝑥 ∈ 𝐴))
11	10	baibr 927	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑋 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ (𝑋 ∩ 𝐴)))
12	11	dcbid 845	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑋 → (DECID 𝑥 ∈ 𝐴 ↔ DECID 𝑥 ∈ (𝑋 ∩ 𝐴)))
13	12	ralbiia 2546	. . . . . . . . . . 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 7114	. . . . . . . . . 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 2267	. . . . . . . 8 ⊢ (𝜑 → ((𝑋 ∩ 𝐴) ∪ (𝑋 ∖ 𝐴)) = 𝑋)
18	17	reseq2d 5013	. . . . . . 7 ⊢ (𝜑 → (𝐹 ↾ ((𝑋 ∩ 𝐴) ∪ (𝑋 ∖ 𝐴))) = (𝐹 ↾ 𝑋))
19		ssidd 3248	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ⊆ 𝑋)
20	19	resmptd 5064	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ if(𝑥 ∈ 𝐴, 1o, ∅)) ↾ 𝑋) = (𝑥 ∈ 𝑋 ↦ if(𝑥 ∈ 𝐴, 1o, ∅)))
21	1	reseq1d 5012	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ↾ 𝑋) = ((𝑥 ∈ 𝑋 ↦ if(𝑥 ∈ 𝐴, 1o, ∅)) ↾ 𝑋))
22	20, 21, 1	3eqtr4d 2274	. . . . . . 7 ⊢ (𝜑 → (𝐹 ↾ 𝑋) = 𝐹)
23	18, 22	eqtrd 2264	. . . . . 6 ⊢ (𝜑 → (𝐹 ↾ ((𝑋 ∩ 𝐴) ∪ (𝑋 ∖ 𝐴))) = 𝐹)
24	23, 17	feq12d 5472	. . . . 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 841   = wceq 1397   ∈ wcel 2202  ∀wral 2510   ∖ cdif 3197   ∪ cun 3198   ∩ cin 3199   ⊆ wss 3200  ∅c0 3494  ifcif 3605  𝒫 cpw 3652   ↦ cmpt 4150   ↾ cres 4727  ⟶wf 5322  ‘cfv 5326  1_oc1o 6574  2_oc2o 6575

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-1o 6581  df-2o 6582

This theorem is referenced by: (None)