Theorem bj-charfundcALT 15914

Description: Alternate proof of bj-charfundc 15913. It was expected to be much shorter since it uses bj-charfun 15912 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 15912	. 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 3414	. . . . . . . . . . . 12 ⊢ (𝑋 ∖ (𝑋 ∩ 𝐴)) = (𝑋 ∖ 𝐴)
4	3	eqcomi 2210	. . . . . . . . . . 11 ⊢ (𝑋 ∖ 𝐴) = (𝑋 ∖ (𝑋 ∩ 𝐴))
5	4	a1i 9	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 ∖ 𝐴) = (𝑋 ∖ (𝑋 ∩ 𝐴)))
6	5	uneq2d 3331	. . . . . . . . 9 ⊢ (𝜑 → ((𝑋 ∩ 𝐴) ∪ (𝑋 ∖ 𝐴)) = ((𝑋 ∩ 𝐴) ∪ (𝑋 ∖ (𝑋 ∩ 𝐴))))
7		inss1 3397	. . . . . . . . . . 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 3360	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (𝑋 ∩ 𝐴) ↔ (𝑥 ∈ 𝑋 ∧ 𝑥 ∈ 𝐴))
11	10	baibr 922	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑋 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ (𝑋 ∩ 𝐴)))
12	11	dcbid 840	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑋 → (DECID 𝑥 ∈ 𝐴 ↔ DECID 𝑥 ∈ (𝑋 ∩ 𝐴)))
13	12	ralbiia 2521	. . . . . . . . . . 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 7041	. . . . . . . . . 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 2242	. . . . . . . 8 ⊢ (𝜑 → ((𝑋 ∩ 𝐴) ∪ (𝑋 ∖ 𝐴)) = 𝑋)
18	17	reseq2d 4973	. . . . . . 7 ⊢ (𝜑 → (𝐹 ↾ ((𝑋 ∩ 𝐴) ∪ (𝑋 ∖ 𝐴))) = (𝐹 ↾ 𝑋))
19		ssidd 3218	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ⊆ 𝑋)
20	19	resmptd 5024	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ if(𝑥 ∈ 𝐴, 1o, ∅)) ↾ 𝑋) = (𝑥 ∈ 𝑋 ↦ if(𝑥 ∈ 𝐴, 1o, ∅)))
21	1	reseq1d 4972	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ↾ 𝑋) = ((𝑥 ∈ 𝑋 ↦ if(𝑥 ∈ 𝐴, 1o, ∅)) ↾ 𝑋))
22	20, 21, 1	3eqtr4d 2249	. . . . . . 7 ⊢ (𝜑 → (𝐹 ↾ 𝑋) = 𝐹)
23	18, 22	eqtrd 2239	. . . . . 6 ⊢ (𝜑 → (𝐹 ↾ ((𝑋 ∩ 𝐴) ∪ (𝑋 ∖ 𝐴))) = 𝐹)
24	23, 17	feq12d 5430	. . . . 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 836   = wceq 1373   ∈ wcel 2177  ∀wral 2485   ∖ cdif 3167   ∪ cun 3168   ∩ cin 3169   ⊆ wss 3170  ∅c0 3464  ifcif 3575  𝒫 cpw 3621   ↦ cmpt 4116   ↾ cres 4690  ⟶wf 5281  ‘cfv 5285  1_oc1o 6513  2_oc2o 6514

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 2179  ax-14 2180  ax-ext 2188  ax-sep 4173  ax-nul 4181  ax-pow 4229  ax-pr 4264  ax-un 4493

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 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-if 3576  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-br 4055  df-opab 4117  df-mpt 4118  df-tr 4154  df-id 4353  df-iord 4426  df-on 4428  df-suc 4431  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-iota 5246  df-fun 5287  df-fn 5288  df-f 5289  df-fv 5293  df-1o 6520  df-2o 6521

This theorem is referenced by: (None)