Theorem bj-charfundcALT 16835

Description: Alternate proof of bj-charfundc 16834. It was expected to be much shorter since it uses bj-charfun 16833 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 16833	. 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 3468	. . . . . . . . . . . 12 ⊢ (𝑋 ∖ (𝑋 ∩ 𝐴)) = (𝑋 ∖ 𝐴)
4	3	eqcomi 2242	. . . . . . . . . . 11 ⊢ (𝑋 ∖ 𝐴) = (𝑋 ∖ (𝑋 ∩ 𝐴))
5	4	a1i 9	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 ∖ 𝐴) = (𝑋 ∖ (𝑋 ∩ 𝐴)))
6	5	uneq2d 3383	. . . . . . . . 9 ⊢ (𝜑 → ((𝑋 ∩ 𝐴) ∪ (𝑋 ∖ 𝐴)) = ((𝑋 ∩ 𝐴) ∪ (𝑋 ∖ (𝑋 ∩ 𝐴))))
7		inss1 3451	. . . . . . . . . . 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 3412	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (𝑋 ∩ 𝐴) ↔ (𝑥 ∈ 𝑋 ∧ 𝑥 ∈ 𝐴))
11	10	baibr 932	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑋 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ (𝑋 ∩ 𝐴)))
12	11	dcbid 850	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑋 → (DECID 𝑥 ∈ 𝐴 ↔ DECID 𝑥 ∈ (𝑋 ∩ 𝐴)))
13	12	ralbiia 2564	. . . . . . . . . . 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 7230	. . . . . . . . . 10 ⊢ (𝑋 = ((𝑋 ∩ 𝐴) ∪ (𝑋 ∖ (𝑋 ∩ 𝐴))) ↔ ((𝑋 ∩ 𝐴) ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝑋 DECID 𝑥 ∈ (𝑋 ∩ 𝐴)))
16	8, 14, 15	sylanbrc 421	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 = ((𝑋 ∩ 𝐴) ∪ (𝑋 ∖ (𝑋 ∩ 𝐴))))
17	6, 16	eqtr4d 2274	. . . . . . . 8 ⊢ (𝜑 → ((𝑋 ∩ 𝐴) ∪ (𝑋 ∖ 𝐴)) = 𝑋)
18	17	reseq2d 5063	. . . . . . 7 ⊢ (𝜑 → (𝐹 ↾ ((𝑋 ∩ 𝐴) ∪ (𝑋 ∖ 𝐴))) = (𝐹 ↾ 𝑋))
19		ssidd 3269	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ⊆ 𝑋)
20	19	resmptd 5114	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ if(𝑥 ∈ 𝐴, 1o, ∅)) ↾ 𝑋) = (𝑥 ∈ 𝑋 ↦ if(𝑥 ∈ 𝐴, 1o, ∅)))
21	1	reseq1d 5062	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ↾ 𝑋) = ((𝑥 ∈ 𝑋 ↦ if(𝑥 ∈ 𝐴, 1o, ∅)) ↾ 𝑋))
22	20, 21, 1	3eqtr4d 2281	. . . . . . 7 ⊢ (𝜑 → (𝐹 ↾ 𝑋) = 𝐹)
23	18, 22	eqtrd 2271	. . . . . 6 ⊢ (𝜑 → (𝐹 ↾ ((𝑋 ∩ 𝐴) ∪ (𝑋 ∖ 𝐴))) = 𝐹)
24	23, 17	feq12d 5523	. . . . 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) = ∅)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104  DECID wdc 846   = wceq 1402   ∈ wcel 2209  ∀wral 2528   ∖ cdif 3217   ∪ cun 3218   ∩ cin 3219   ⊆ wss 3220  ∅c0 3520  ifcif 3638  𝒫 cpw 3688   ↦ cmpt 4192   ↾ cres 4776  ⟶wf 5373  ‘cfv 5377  1_oc1o 6680  2_oc2o 6681

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578

This proof depends on definitions:  df-bi 117  df-dc 847  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-suc 4516  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385  df-1o 6687  df-2o 6688

This theorem is used by: (None)