Theorem bj-charfunr 16580

Description: If a class 𝐴 has a "weak" characteristic function on a class 𝑋, then negated membership in 𝐴 is decidable (in other words, membership in 𝐴 is testable) in 𝑋.

The hypothesis imposes that 𝑋 be a set. As usual, it could be formulated as ⊢ (𝜑 → (𝐹:𝑋⟶ω ∧ ...)) to deal with general classes, but that extra generality would not make the theorem much more useful.

The theorem would still hold if the codomain of 𝑓 were any class with testable equality to the point where (𝑋 ∖ 𝐴) is sent. (Contributed by BJ, 6-Aug-2024.)

Hypothesis

Ref	Expression
bj-charfunr.1	⊢ (𝜑 → ∃𝑓 ∈ (ω ↑𝑚 𝑋)(∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) ≠ ∅ ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅))

Assertion

Ref	Expression
bj-charfunr	⊢ (𝜑 → ∀𝑥 ∈ 𝑋 DECID ¬ 𝑥 ∈ 𝐴)

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

Allowed substitution hints:   𝐴(𝑥)   𝑋(𝑥)

Proof of Theorem bj-charfunr

Step	Hyp	Ref	Expression
1		bj-charfunr.1	. . . . 5 ⊢ (𝜑 → ∃𝑓 ∈ (ω ↑𝑚 𝑋)(∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) ≠ ∅ ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅))
2		elmapi 6904	. . . . . . . . . 10 ⊢ (𝑓 ∈ (ω ↑𝑚 𝑋) → 𝑓:𝑋⟶ω)
3		ffvelcdm 5810	. . . . . . . . . . 11 ⊢ ((𝑓:𝑋⟶ω ∧ 𝑥 ∈ 𝑋) → (𝑓‘𝑥) ∈ ω)
4	3	ex 115	. . . . . . . . . 10 ⊢ (𝑓:𝑋⟶ω → (𝑥 ∈ 𝑋 → (𝑓‘𝑥) ∈ ω))
5	2, 4	syl 14	. . . . . . . . 9 ⊢ (𝑓 ∈ (ω ↑𝑚 𝑋) → (𝑥 ∈ 𝑋 → (𝑓‘𝑥) ∈ ω))
6		0elnn 4741	. . . . . . . . . 10 ⊢ ((𝑓‘𝑥) ∈ ω → ((𝑓‘𝑥) = ∅ ∨ ∅ ∈ (𝑓‘𝑥)))
7		nn0eln0 4742	. . . . . . . . . . 11 ⊢ ((𝑓‘𝑥) ∈ ω → (∅ ∈ (𝑓‘𝑥) ↔ (𝑓‘𝑥) ≠ ∅))
8	7	orbi2d 798	. . . . . . . . . 10 ⊢ ((𝑓‘𝑥) ∈ ω → (((𝑓‘𝑥) = ∅ ∨ ∅ ∈ (𝑓‘𝑥)) ↔ ((𝑓‘𝑥) = ∅ ∨ (𝑓‘𝑥) ≠ ∅)))
9	6, 8	mpbid 147	. . . . . . . . 9 ⊢ ((𝑓‘𝑥) ∈ ω → ((𝑓‘𝑥) = ∅ ∨ (𝑓‘𝑥) ≠ ∅))
10	5, 9	syl6 33	. . . . . . . 8 ⊢ (𝑓 ∈ (ω ↑𝑚 𝑋) → (𝑥 ∈ 𝑋 → ((𝑓‘𝑥) = ∅ ∨ (𝑓‘𝑥) ≠ ∅)))
11	10	adantr 276	. . . . . . 7 ⊢ ((𝑓 ∈ (ω ↑𝑚 𝑋) ∧ (∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) ≠ ∅ ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅)) → (𝑥 ∈ 𝑋 → ((𝑓‘𝑥) = ∅ ∨ (𝑓‘𝑥) ≠ ∅)))
12		elin 3402	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (𝑋 ∩ 𝐴) ↔ (𝑥 ∈ 𝑋 ∧ 𝑥 ∈ 𝐴))
13		rsp 2589	. . . . . . . . . . . . . . 15 ⊢ (∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) ≠ ∅ → (𝑥 ∈ (𝑋 ∩ 𝐴) → (𝑓‘𝑥) ≠ ∅))
14	12, 13	biimtrrid 153	. . . . . . . . . . . . . 14 ⊢ (∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) ≠ ∅ → ((𝑥 ∈ 𝑋 ∧ 𝑥 ∈ 𝐴) → (𝑓‘𝑥) ≠ ∅))
15	14	expd 258	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) ≠ ∅ → (𝑥 ∈ 𝑋 → (𝑥 ∈ 𝐴 → (𝑓‘𝑥) ≠ ∅)))
16	15	adantr 276	. . . . . . . . . . . 12 ⊢ ((∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) ≠ ∅ ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅) → (𝑥 ∈ 𝑋 → (𝑥 ∈ 𝐴 → (𝑓‘𝑥) ≠ ∅)))
17	16	imp 124	. . . . . . . . . . 11 ⊢ (((∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) ≠ ∅ ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅) ∧ 𝑥 ∈ 𝑋) → (𝑥 ∈ 𝐴 → (𝑓‘𝑥) ≠ ∅))
18	17	necon2bd 2470	. . . . . . . . . 10 ⊢ (((∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) ≠ ∅ ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅) ∧ 𝑥 ∈ 𝑋) → ((𝑓‘𝑥) = ∅ → ¬ 𝑥 ∈ 𝐴))
19		eldif 3220	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (𝑋 ∖ 𝐴) ↔ (𝑥 ∈ 𝑋 ∧ ¬ 𝑥 ∈ 𝐴))
20		rsp 2589	. . . . . . . . . . . . . . 15 ⊢ (∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅ → (𝑥 ∈ (𝑋 ∖ 𝐴) → (𝑓‘𝑥) = ∅))
21	19, 20	biimtrrid 153	. . . . . . . . . . . . . 14 ⊢ (∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅ → ((𝑥 ∈ 𝑋 ∧ ¬ 𝑥 ∈ 𝐴) → (𝑓‘𝑥) = ∅))
22	21	expd 258	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅ → (𝑥 ∈ 𝑋 → (¬ 𝑥 ∈ 𝐴 → (𝑓‘𝑥) = ∅)))
23	22	adantl 277	. . . . . . . . . . . 12 ⊢ ((∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) ≠ ∅ ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅) → (𝑥 ∈ 𝑋 → (¬ 𝑥 ∈ 𝐴 → (𝑓‘𝑥) = ∅)))
24	23	imp 124	. . . . . . . . . . 11 ⊢ (((∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) ≠ ∅ ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅) ∧ 𝑥 ∈ 𝑋) → (¬ 𝑥 ∈ 𝐴 → (𝑓‘𝑥) = ∅))
25	24	necon3ad 2454	. . . . . . . . . 10 ⊢ (((∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) ≠ ∅ ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅) ∧ 𝑥 ∈ 𝑋) → ((𝑓‘𝑥) ≠ ∅ → ¬ ¬ 𝑥 ∈ 𝐴))
26	18, 25	orim12d 794	. . . . . . . . 9 ⊢ (((∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) ≠ ∅ ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅) ∧ 𝑥 ∈ 𝑋) → (((𝑓‘𝑥) = ∅ ∨ (𝑓‘𝑥) ≠ ∅) → (¬ 𝑥 ∈ 𝐴 ∨ ¬ ¬ 𝑥 ∈ 𝐴)))
27	26	ex 115	. . . . . . . 8 ⊢ ((∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) ≠ ∅ ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅) → (𝑥 ∈ 𝑋 → (((𝑓‘𝑥) = ∅ ∨ (𝑓‘𝑥) ≠ ∅) → (¬ 𝑥 ∈ 𝐴 ∨ ¬ ¬ 𝑥 ∈ 𝐴))))
28	27	adantl 277	. . . . . . 7 ⊢ ((𝑓 ∈ (ω ↑𝑚 𝑋) ∧ (∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) ≠ ∅ ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅)) → (𝑥 ∈ 𝑋 → (((𝑓‘𝑥) = ∅ ∨ (𝑓‘𝑥) ≠ ∅) → (¬ 𝑥 ∈ 𝐴 ∨ ¬ ¬ 𝑥 ∈ 𝐴))))
29	11, 28	mpdd 41	. . . . . 6 ⊢ ((𝑓 ∈ (ω ↑𝑚 𝑋) ∧ (∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) ≠ ∅ ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅)) → (𝑥 ∈ 𝑋 → (¬ 𝑥 ∈ 𝐴 ∨ ¬ ¬ 𝑥 ∈ 𝐴)))
30	29	adantl 277	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (ω ↑𝑚 𝑋) ∧ (∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) ≠ ∅ ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅))) → (𝑥 ∈ 𝑋 → (¬ 𝑥 ∈ 𝐴 ∨ ¬ ¬ 𝑥 ∈ 𝐴)))
31	1, 30	rexlimddv 2665	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋 → (¬ 𝑥 ∈ 𝐴 ∨ ¬ ¬ 𝑥 ∈ 𝐴)))
32	31	imp 124	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (¬ 𝑥 ∈ 𝐴 ∨ ¬ ¬ 𝑥 ∈ 𝐴))
33		df-dc 843	. . 3 ⊢ (DECID ¬ 𝑥 ∈ 𝐴 ↔ (¬ 𝑥 ∈ 𝐴 ∨ ¬ ¬ 𝑥 ∈ 𝐴))
34	32, 33	sylibr 134	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → DECID ¬ 𝑥 ∈ 𝐴)
35	34	ralrimiva 2615	1 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 DECID ¬ 𝑥 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 716  DECID wdc 842   = wceq 1398   ∈ wcel 2203   ≠ wne 2412  ∀wral 2520  ∃wrex 2521   ∖ cdif 3208   ∩ cin 3210  ∅c0 3508  ωcom 4712  ⟶wf 5348  ‘cfv 5352  (class class class)co 6050   ↑_𝑚 cmap 6882

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  ax-setind 4659  ax-iinf 4710

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-ne 2413  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-id 4414  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-map 6884

This theorem is referenced by:  bj-charfunbi  16581