Theorem bj-charfunr 13845

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 6648	. . . . . . . . . 10 ⊢ (𝑓 ∈ (ω ↑𝑚 𝑋) → 𝑓:𝑋⟶ω)
3		ffvelrn 5629	. . . . . . . . . . 11 ⊢ ((𝑓:𝑋⟶ω ∧ 𝑥 ∈ 𝑋) → (𝑓‘𝑥) ∈ ω)
4	3	ex 114	. . . . . . . . . 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 4603	. . . . . . . . . 10 ⊢ ((𝑓‘𝑥) ∈ ω → ((𝑓‘𝑥) = ∅ ∨ ∅ ∈ (𝑓‘𝑥)))
7		nn0eln0 4604	. . . . . . . . . . 11 ⊢ ((𝑓‘𝑥) ∈ ω → (∅ ∈ (𝑓‘𝑥) ↔ (𝑓‘𝑥) ≠ ∅))
8	7	orbi2d 785	. . . . . . . . . 10 ⊢ ((𝑓‘𝑥) ∈ ω → (((𝑓‘𝑥) = ∅ ∨ ∅ ∈ (𝑓‘𝑥)) ↔ ((𝑓‘𝑥) = ∅ ∨ (𝑓‘𝑥) ≠ ∅)))
9	6, 8	mpbid 146	. . . . . . . . 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 274	. . . . . . 7 ⊢ ((𝑓 ∈ (ω ↑𝑚 𝑋) ∧ (∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) ≠ ∅ ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅)) → (𝑥 ∈ 𝑋 → ((𝑓‘𝑥) = ∅ ∨ (𝑓‘𝑥) ≠ ∅)))
12		elin 3310	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (𝑋 ∩ 𝐴) ↔ (𝑥 ∈ 𝑋 ∧ 𝑥 ∈ 𝐴))
13		rsp 2517	. . . . . . . . . . . . . . 15 ⊢ (∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) ≠ ∅ → (𝑥 ∈ (𝑋 ∩ 𝐴) → (𝑓‘𝑥) ≠ ∅))
14	12, 13	syl5bir 152	. . . . . . . . . . . . . 14 ⊢ (∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) ≠ ∅ → ((𝑥 ∈ 𝑋 ∧ 𝑥 ∈ 𝐴) → (𝑓‘𝑥) ≠ ∅))
15	14	expd 256	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) ≠ ∅ → (𝑥 ∈ 𝑋 → (𝑥 ∈ 𝐴 → (𝑓‘𝑥) ≠ ∅)))
16	15	adantr 274	. . . . . . . . . . . 12 ⊢ ((∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) ≠ ∅ ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅) → (𝑥 ∈ 𝑋 → (𝑥 ∈ 𝐴 → (𝑓‘𝑥) ≠ ∅)))
17	16	imp 123	. . . . . . . . . . 11 ⊢ (((∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) ≠ ∅ ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅) ∧ 𝑥 ∈ 𝑋) → (𝑥 ∈ 𝐴 → (𝑓‘𝑥) ≠ ∅))
18	17	necon2bd 2398	. . . . . . . . . 10 ⊢ (((∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) ≠ ∅ ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅) ∧ 𝑥 ∈ 𝑋) → ((𝑓‘𝑥) = ∅ → ¬ 𝑥 ∈ 𝐴))
19		eldif 3130	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (𝑋 ∖ 𝐴) ↔ (𝑥 ∈ 𝑋 ∧ ¬ 𝑥 ∈ 𝐴))
20		rsp 2517	. . . . . . . . . . . . . . 15 ⊢ (∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅ → (𝑥 ∈ (𝑋 ∖ 𝐴) → (𝑓‘𝑥) = ∅))
21	19, 20	syl5bir 152	. . . . . . . . . . . . . 14 ⊢ (∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅ → ((𝑥 ∈ 𝑋 ∧ ¬ 𝑥 ∈ 𝐴) → (𝑓‘𝑥) = ∅))
22	21	expd 256	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅ → (𝑥 ∈ 𝑋 → (¬ 𝑥 ∈ 𝐴 → (𝑓‘𝑥) = ∅)))
23	22	adantl 275	. . . . . . . . . . . 12 ⊢ ((∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) ≠ ∅ ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅) → (𝑥 ∈ 𝑋 → (¬ 𝑥 ∈ 𝐴 → (𝑓‘𝑥) = ∅)))
24	23	imp 123	. . . . . . . . . . 11 ⊢ (((∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) ≠ ∅ ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅) ∧ 𝑥 ∈ 𝑋) → (¬ 𝑥 ∈ 𝐴 → (𝑓‘𝑥) = ∅))
25	24	necon3ad 2382	. . . . . . . . . 10 ⊢ (((∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) ≠ ∅ ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅) ∧ 𝑥 ∈ 𝑋) → ((𝑓‘𝑥) ≠ ∅ → ¬ ¬ 𝑥 ∈ 𝐴))
26	18, 25	orim12d 781	. . . . . . . . 9 ⊢ (((∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) ≠ ∅ ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅) ∧ 𝑥 ∈ 𝑋) → (((𝑓‘𝑥) = ∅ ∨ (𝑓‘𝑥) ≠ ∅) → (¬ 𝑥 ∈ 𝐴 ∨ ¬ ¬ 𝑥 ∈ 𝐴)))
27	26	ex 114	. . . . . . . 8 ⊢ ((∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) ≠ ∅ ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅) → (𝑥 ∈ 𝑋 → (((𝑓‘𝑥) = ∅ ∨ (𝑓‘𝑥) ≠ ∅) → (¬ 𝑥 ∈ 𝐴 ∨ ¬ ¬ 𝑥 ∈ 𝐴))))
28	27	adantl 275	. . . . . . 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 275	. . . . 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 2592	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋 → (¬ 𝑥 ∈ 𝐴 ∨ ¬ ¬ 𝑥 ∈ 𝐴)))
32	31	imp 123	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (¬ 𝑥 ∈ 𝐴 ∨ ¬ ¬ 𝑥 ∈ 𝐴))
33		df-dc 830	. . 3 ⊢ (DECID ¬ 𝑥 ∈ 𝐴 ↔ (¬ 𝑥 ∈ 𝐴 ∨ ¬ ¬ 𝑥 ∈ 𝐴))
34	32, 33	sylibr 133	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → DECID ¬ 𝑥 ∈ 𝐴)
35	34	ralrimiva 2543	1 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 DECID ¬ 𝑥 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∨ wo 703  DECID wdc 829   = wceq 1348   ∈ wcel 2141   ≠ wne 2340  ∀wral 2448  ∃wrex 2449   ∖ cdif 3118   ∩ cin 3120  ∅c0 3414  ωcom 4574  ⟶wf 5194  ‘cfv 5198  (class class class)co 5853   ↑_𝑚 cmap 6626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-id 4278  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-map 6628

This theorem is referenced by:  bj-charfunbi  13846