Theorem bj-charfundcALT 15745

Description: Alternate proof of bj-charfundc 15744. It was expected to be much shorter since it uses bj-charfun 15743 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 15743	. 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 3410	. . . . . . . . . . . 12 ⊢ (𝑋 ∖ (𝑋 ∩ 𝐴)) = (𝑋 ∖ 𝐴)
4	3	eqcomi 2209	. . . . . . . . . . 11 ⊢ (𝑋 ∖ 𝐴) = (𝑋 ∖ (𝑋 ∩ 𝐴))
5	4	a1i 9	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 ∖ 𝐴) = (𝑋 ∖ (𝑋 ∩ 𝐴)))
6	5	uneq2d 3327	. . . . . . . . 9 ⊢ (𝜑 → ((𝑋 ∩ 𝐴) ∪ (𝑋 ∖ 𝐴)) = ((𝑋 ∩ 𝐴) ∪ (𝑋 ∖ (𝑋 ∩ 𝐴))))
7		inss1 3393	. . . . . . . . . . 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 3356	. . . . . . . . . . . . . 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 2520	. . . . . . . . . . 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 7020	. . . . . . . . . 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 2241	. . . . . . . 8 ⊢ (𝜑 → ((𝑋 ∩ 𝐴) ∪ (𝑋 ∖ 𝐴)) = 𝑋)
18	17	reseq2d 4959	. . . . . . 7 ⊢ (𝜑 → (𝐹 ↾ ((𝑋 ∩ 𝐴) ∪ (𝑋 ∖ 𝐴))) = (𝐹 ↾ 𝑋))
19		ssidd 3214	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ⊆ 𝑋)
20	19	resmptd 5010	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ if(𝑥 ∈ 𝐴, 1o, ∅)) ↾ 𝑋) = (𝑥 ∈ 𝑋 ↦ if(𝑥 ∈ 𝐴, 1o, ∅)))
21	1	reseq1d 4958	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ↾ 𝑋) = ((𝑥 ∈ 𝑋 ↦ if(𝑥 ∈ 𝐴, 1o, ∅)) ↾ 𝑋))
22	20, 21, 1	3eqtr4d 2248	. . . . . . 7 ⊢ (𝜑 → (𝐹 ↾ 𝑋) = 𝐹)
23	18, 22	eqtrd 2238	. . . . . 6 ⊢ (𝜑 → (𝐹 ↾ ((𝑋 ∩ 𝐴) ∪ (𝑋 ∖ 𝐴))) = 𝐹)
24	23, 17	feq12d 5415	. . . . 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 2176  ∀wral 2484   ∖ cdif 3163   ∪ cun 3164   ∩ cin 3165   ⊆ wss 3166  ∅c0 3460  ifcif 3571  𝒫 cpw 3616   ↦ cmpt 4105   ↾ cres 4677  ⟶wf 5267  ‘cfv 5271  1_oc1o 6495  2_oc2o 6496

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-id 4340  df-iord 4413  df-on 4415  df-suc 4418  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-1o 6502  df-2o 6503

This theorem is referenced by: (None)