Theorem bj-charfunr 14498

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 6669	. . . . . . . . . 10 ⊢ (𝑓 ∈ (ω ↑𝑚 𝑋) → 𝑓:𝑋⟶ω)
3		ffvelcdm 5649	. . . . . . . . . . 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 4618	. . . . . . . . . 10 ⊢ ((𝑓‘𝑥) ∈ ω → ((𝑓‘𝑥) = ∅ ∨ ∅ ∈ (𝑓‘𝑥)))
7		nn0eln0 4619	. . . . . . . . . . 11 ⊢ ((𝑓‘𝑥) ∈ ω → (∅ ∈ (𝑓‘𝑥) ↔ (𝑓‘𝑥) ≠ ∅))
8	7	orbi2d 790	. . . . . . . . . 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 3318	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (𝑋 ∩ 𝐴) ↔ (𝑥 ∈ 𝑋 ∧ 𝑥 ∈ 𝐴))
13		rsp 2524	. . . . . . . . . . . . . . 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 2405	. . . . . . . . . 10 ⊢ (((∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) ≠ ∅ ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅) ∧ 𝑥 ∈ 𝑋) → ((𝑓‘𝑥) = ∅ → ¬ 𝑥 ∈ 𝐴))
19		eldif 3138	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (𝑋 ∖ 𝐴) ↔ (𝑥 ∈ 𝑋 ∧ ¬ 𝑥 ∈ 𝐴))
20		rsp 2524	. . . . . . . . . . . . . . 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 2389	. . . . . . . . . 10 ⊢ (((∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) ≠ ∅ ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅) ∧ 𝑥 ∈ 𝑋) → ((𝑓‘𝑥) ≠ ∅ → ¬ ¬ 𝑥 ∈ 𝐴))
26	18, 25	orim12d 786	. . . . . . . . 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 2599	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋 → (¬ 𝑥 ∈ 𝐴 ∨ ¬ ¬ 𝑥 ∈ 𝐴)))
32	31	imp 124	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (¬ 𝑥 ∈ 𝐴 ∨ ¬ ¬ 𝑥 ∈ 𝐴))
33		df-dc 835	. . 3 ⊢ (DECID ¬ 𝑥 ∈ 𝐴 ↔ (¬ 𝑥 ∈ 𝐴 ∨ ¬ ¬ 𝑥 ∈ 𝐴))
34	32, 33	sylibr 134	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → DECID ¬ 𝑥 ∈ 𝐴)
35	34	ralrimiva 2550	1 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 DECID ¬ 𝑥 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 708  DECID wdc 834   = wceq 1353   ∈ wcel 2148   ≠ wne 2347  ∀wral 2455  ∃wrex 2456   ∖ cdif 3126   ∩ cin 3128  ∅c0 3422  ωcom 4589  ⟶wf 5212  ‘cfv 5216  (class class class)co 5874   ↑_𝑚 cmap 6647

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4004  df-opab 4065  df-id 4293  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-map 6649

This theorem is referenced by:  bj-charfunbi  14499