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Theorem bj-charfunr 14565
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 (𝑋𝐴) is sent. (Contributed by BJ, 6-Aug-2024.)

Hypothesis
Ref Expression
bj-charfunr.1 (𝜑 → ∃𝑓 ∈ (ω ↑𝑚 𝑋)(∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) ≠ ∅ ∧ ∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = ∅))
Assertion
Ref Expression
bj-charfunr (𝜑 → ∀𝑥𝑋 DECID ¬ 𝑥𝐴)
Distinct variable groups:   𝐴,𝑓   𝑓,𝑋   𝜑,𝑓,𝑥
Allowed substitution hints:   𝐴(𝑥)   𝑋(𝑥)

Proof of Theorem bj-charfunr
StepHypRef Expression
1 bj-charfunr.1 . . . . 5 (𝜑 → ∃𝑓 ∈ (ω ↑𝑚 𝑋)(∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) ≠ ∅ ∧ ∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = ∅))
2 elmapi 6670 . . . . . . . . . 10 (𝑓 ∈ (ω ↑𝑚 𝑋) → 𝑓:𝑋⟶ω)
3 ffvelcdm 5650 . . . . . . . . . . 11 ((𝑓:𝑋⟶ω ∧ 𝑥𝑋) → (𝑓𝑥) ∈ ω)
43ex 115 . . . . . . . . . 10 (𝑓:𝑋⟶ω → (𝑥𝑋 → (𝑓𝑥) ∈ ω))
52, 4syl 14 . . . . . . . . 9 (𝑓 ∈ (ω ↑𝑚 𝑋) → (𝑥𝑋 → (𝑓𝑥) ∈ ω))
6 0elnn 4619 . . . . . . . . . 10 ((𝑓𝑥) ∈ ω → ((𝑓𝑥) = ∅ ∨ ∅ ∈ (𝑓𝑥)))
7 nn0eln0 4620 . . . . . . . . . . 11 ((𝑓𝑥) ∈ ω → (∅ ∈ (𝑓𝑥) ↔ (𝑓𝑥) ≠ ∅))
87orbi2d 790 . . . . . . . . . 10 ((𝑓𝑥) ∈ ω → (((𝑓𝑥) = ∅ ∨ ∅ ∈ (𝑓𝑥)) ↔ ((𝑓𝑥) = ∅ ∨ (𝑓𝑥) ≠ ∅)))
96, 8mpbid 147 . . . . . . . . 9 ((𝑓𝑥) ∈ ω → ((𝑓𝑥) = ∅ ∨ (𝑓𝑥) ≠ ∅))
105, 9syl6 33 . . . . . . . 8 (𝑓 ∈ (ω ↑𝑚 𝑋) → (𝑥𝑋 → ((𝑓𝑥) = ∅ ∨ (𝑓𝑥) ≠ ∅)))
1110adantr 276 . . . . . . 7 ((𝑓 ∈ (ω ↑𝑚 𝑋) ∧ (∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) ≠ ∅ ∧ ∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = ∅)) → (𝑥𝑋 → ((𝑓𝑥) = ∅ ∨ (𝑓𝑥) ≠ ∅)))
12 elin 3319 . . . . . . . . . . . . . . 15 (𝑥 ∈ (𝑋𝐴) ↔ (𝑥𝑋𝑥𝐴))
13 rsp 2524 . . . . . . . . . . . . . . 15 (∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) ≠ ∅ → (𝑥 ∈ (𝑋𝐴) → (𝑓𝑥) ≠ ∅))
1412, 13biimtrrid 153 . . . . . . . . . . . . . 14 (∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) ≠ ∅ → ((𝑥𝑋𝑥𝐴) → (𝑓𝑥) ≠ ∅))
1514expd 258 . . . . . . . . . . . . 13 (∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) ≠ ∅ → (𝑥𝑋 → (𝑥𝐴 → (𝑓𝑥) ≠ ∅)))
1615adantr 276 . . . . . . . . . . . 12 ((∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) ≠ ∅ ∧ ∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = ∅) → (𝑥𝑋 → (𝑥𝐴 → (𝑓𝑥) ≠ ∅)))
1716imp 124 . . . . . . . . . . 11 (((∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) ≠ ∅ ∧ ∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = ∅) ∧ 𝑥𝑋) → (𝑥𝐴 → (𝑓𝑥) ≠ ∅))
1817necon2bd 2405 . . . . . . . . . 10 (((∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) ≠ ∅ ∧ ∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = ∅) ∧ 𝑥𝑋) → ((𝑓𝑥) = ∅ → ¬ 𝑥𝐴))
19 eldif 3139 . . . . . . . . . . . . . . 15 (𝑥 ∈ (𝑋𝐴) ↔ (𝑥𝑋 ∧ ¬ 𝑥𝐴))
20 rsp 2524 . . . . . . . . . . . . . . 15 (∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = ∅ → (𝑥 ∈ (𝑋𝐴) → (𝑓𝑥) = ∅))
2119, 20biimtrrid 153 . . . . . . . . . . . . . 14 (∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = ∅ → ((𝑥𝑋 ∧ ¬ 𝑥𝐴) → (𝑓𝑥) = ∅))
2221expd 258 . . . . . . . . . . . . 13 (∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = ∅ → (𝑥𝑋 → (¬ 𝑥𝐴 → (𝑓𝑥) = ∅)))
2322adantl 277 . . . . . . . . . . . 12 ((∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) ≠ ∅ ∧ ∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = ∅) → (𝑥𝑋 → (¬ 𝑥𝐴 → (𝑓𝑥) = ∅)))
2423imp 124 . . . . . . . . . . 11 (((∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) ≠ ∅ ∧ ∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = ∅) ∧ 𝑥𝑋) → (¬ 𝑥𝐴 → (𝑓𝑥) = ∅))
2524necon3ad 2389 . . . . . . . . . 10 (((∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) ≠ ∅ ∧ ∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = ∅) ∧ 𝑥𝑋) → ((𝑓𝑥) ≠ ∅ → ¬ ¬ 𝑥𝐴))
2618, 25orim12d 786 . . . . . . . . 9 (((∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) ≠ ∅ ∧ ∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = ∅) ∧ 𝑥𝑋) → (((𝑓𝑥) = ∅ ∨ (𝑓𝑥) ≠ ∅) → (¬ 𝑥𝐴 ∨ ¬ ¬ 𝑥𝐴)))
2726ex 115 . . . . . . . 8 ((∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) ≠ ∅ ∧ ∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = ∅) → (𝑥𝑋 → (((𝑓𝑥) = ∅ ∨ (𝑓𝑥) ≠ ∅) → (¬ 𝑥𝐴 ∨ ¬ ¬ 𝑥𝐴))))
2827adantl 277 . . . . . . 7 ((𝑓 ∈ (ω ↑𝑚 𝑋) ∧ (∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) ≠ ∅ ∧ ∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = ∅)) → (𝑥𝑋 → (((𝑓𝑥) = ∅ ∨ (𝑓𝑥) ≠ ∅) → (¬ 𝑥𝐴 ∨ ¬ ¬ 𝑥𝐴))))
2911, 28mpdd 41 . . . . . 6 ((𝑓 ∈ (ω ↑𝑚 𝑋) ∧ (∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) ≠ ∅ ∧ ∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = ∅)) → (𝑥𝑋 → (¬ 𝑥𝐴 ∨ ¬ ¬ 𝑥𝐴)))
3029adantl 277 . . . . 5 ((𝜑 ∧ (𝑓 ∈ (ω ↑𝑚 𝑋) ∧ (∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) ≠ ∅ ∧ ∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = ∅))) → (𝑥𝑋 → (¬ 𝑥𝐴 ∨ ¬ ¬ 𝑥𝐴)))
311, 30rexlimddv 2599 . . . 4 (𝜑 → (𝑥𝑋 → (¬ 𝑥𝐴 ∨ ¬ ¬ 𝑥𝐴)))
3231imp 124 . . 3 ((𝜑𝑥𝑋) → (¬ 𝑥𝐴 ∨ ¬ ¬ 𝑥𝐴))
33 df-dc 835 . . 3 (DECID ¬ 𝑥𝐴 ↔ (¬ 𝑥𝐴 ∨ ¬ ¬ 𝑥𝐴))
3432, 33sylibr 134 . 2 ((𝜑𝑥𝑋) → DECID ¬ 𝑥𝐴)
3534ralrimiva 2550 1 (𝜑 → ∀𝑥𝑋 DECID ¬ 𝑥𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wo 708  DECID wdc 834   = wceq 1353  wcel 2148  wne 2347  wral 2455  wrex 2456  cdif 3127  cin 3129  c0 3423  ωcom 4590  wf 5213  cfv 5217  (class class class)co 5875  𝑚 cmap 6648
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 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588
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 2740  df-sbc 2964  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-br 4005  df-opab 4066  df-id 4294  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-map 6650
This theorem is referenced by:  bj-charfunbi  14566
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