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Theorem bj-charfunbi 13693
Description: In an ambient set 𝑋, if membership in 𝐴 is stable, then it is decidable if and only if 𝐴 has a characteristic function.

This characterization can be applied to singletons when the set 𝑋 has stable equality, which is the case as soon as it has a tight apartness relation. (Contributed by BJ, 6-Aug-2024.)

Hypotheses
Ref Expression
bj-charfunbi.ex (𝜑𝑋𝑉)
bj-charfunbi.st (𝜑 → ∀𝑥𝑋 STAB 𝑥𝐴)
Assertion
Ref Expression
bj-charfunbi (𝜑 → (∀𝑥𝑋 DECID 𝑥𝐴 ↔ ∃𝑓 ∈ (2o𝑚 𝑋)(∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = ∅)))
Distinct variable groups:   𝐴,𝑓,𝑥   𝑓,𝑋,𝑥   𝜑,𝑓,𝑥
Allowed substitution hints:   𝑉(𝑥,𝑓)

Proof of Theorem bj-charfunbi
Dummy variables 𝑔 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1w 2227 . . . . 5 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
21dcbid 828 . . . 4 (𝑥 = 𝑧 → (DECID 𝑥𝐴DECID 𝑧𝐴))
32cbvralvw 2696 . . 3 (∀𝑥𝑋 DECID 𝑥𝐴 ↔ ∀𝑧𝑋 DECID 𝑧𝐴)
4 eleq1w 2227 . . . . . . . . . . . 12 (𝑧 = 𝑥 → (𝑧𝐴𝑥𝐴))
54ifbid 3541 . . . . . . . . . . 11 (𝑧 = 𝑥 → if(𝑧𝐴, 1o, ∅) = if(𝑥𝐴, 1o, ∅))
65cbvmptv 4078 . . . . . . . . . 10 (𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅)) = (𝑥𝑋 ↦ if(𝑥𝐴, 1o, ∅))
76a1i 9 . . . . . . . . 9 ((𝜑 ∧ ∀𝑧𝑋 DECID 𝑧𝐴) → (𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅)) = (𝑥𝑋 ↦ if(𝑥𝐴, 1o, ∅)))
83biimpri 132 . . . . . . . . . 10 (∀𝑧𝑋 DECID 𝑧𝐴 → ∀𝑥𝑋 DECID 𝑥𝐴)
98adantl 275 . . . . . . . . 9 ((𝜑 ∧ ∀𝑧𝑋 DECID 𝑧𝐴) → ∀𝑥𝑋 DECID 𝑥𝐴)
107, 9bj-charfundc 13690 . . . . . . . 8 ((𝜑 ∧ ∀𝑧𝑋 DECID 𝑧𝐴) → ((𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅)):𝑋⟶2o ∧ (∀𝑥 ∈ (𝑋𝐴)((𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅))‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋𝐴)((𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅))‘𝑥) = ∅)))
1110ex 114 . . . . . . 7 (𝜑 → (∀𝑧𝑋 DECID 𝑧𝐴 → ((𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅)):𝑋⟶2o ∧ (∀𝑥 ∈ (𝑋𝐴)((𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅))‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋𝐴)((𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅))‘𝑥) = ∅))))
12 2on 6393 . . . . . . . . . . 11 2o ∈ On
1312a1i 9 . . . . . . . . . 10 (𝜑 → 2o ∈ On)
14 bj-charfunbi.ex . . . . . . . . . 10 (𝜑𝑋𝑉)
1513, 14elmapd 6628 . . . . . . . . 9 (𝜑 → ((𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅)) ∈ (2o𝑚 𝑋) ↔ (𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅)):𝑋⟶2o))
1615biimprd 157 . . . . . . . 8 (𝜑 → ((𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅)):𝑋⟶2o → (𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅)) ∈ (2o𝑚 𝑋)))
1716adantrd 277 . . . . . . 7 (𝜑 → (((𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅)):𝑋⟶2o ∧ (∀𝑥 ∈ (𝑋𝐴)((𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅))‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋𝐴)((𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅))‘𝑥) = ∅)) → (𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅)) ∈ (2o𝑚 𝑋)))
1811, 17syld 45 . . . . . 6 (𝜑 → (∀𝑧𝑋 DECID 𝑧𝐴 → (𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅)) ∈ (2o𝑚 𝑋)))
1918imp 123 . . . . 5 ((𝜑 ∧ ∀𝑧𝑋 DECID 𝑧𝐴) → (𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅)) ∈ (2o𝑚 𝑋))
20 fveq1 5485 . . . . . . . . 9 (𝑓 = (𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅)) → (𝑓𝑥) = ((𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅))‘𝑥))
2120eqeq1d 2174 . . . . . . . 8 (𝑓 = (𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅)) → ((𝑓𝑥) = 1o ↔ ((𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅))‘𝑥) = 1o))
2221ralbidv 2466 . . . . . . 7 (𝑓 = (𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅)) → (∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = 1o ↔ ∀𝑥 ∈ (𝑋𝐴)((𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅))‘𝑥) = 1o))
2320eqeq1d 2174 . . . . . . . 8 (𝑓 = (𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅)) → ((𝑓𝑥) = ∅ ↔ ((𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅))‘𝑥) = ∅))
2423ralbidv 2466 . . . . . . 7 (𝑓 = (𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅)) → (∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = ∅ ↔ ∀𝑥 ∈ (𝑋𝐴)((𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅))‘𝑥) = ∅))
2522, 24anbi12d 465 . . . . . 6 (𝑓 = (𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅)) → ((∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = ∅) ↔ (∀𝑥 ∈ (𝑋𝐴)((𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅))‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋𝐴)((𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅))‘𝑥) = ∅)))
2625adantl 275 . . . . 5 (((𝜑 ∧ ∀𝑧𝑋 DECID 𝑧𝐴) ∧ 𝑓 = (𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅))) → ((∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = ∅) ↔ (∀𝑥 ∈ (𝑋𝐴)((𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅))‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋𝐴)((𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅))‘𝑥) = ∅)))
2710simprd 113 . . . . 5 ((𝜑 ∧ ∀𝑧𝑋 DECID 𝑧𝐴) → (∀𝑥 ∈ (𝑋𝐴)((𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅))‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋𝐴)((𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅))‘𝑥) = ∅))
2819, 26, 27rspcedvd 2836 . . . 4 ((𝜑 ∧ ∀𝑧𝑋 DECID 𝑧𝐴) → ∃𝑓 ∈ (2o𝑚 𝑋)(∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = ∅))
2928ex 114 . . 3 (𝜑 → (∀𝑧𝑋 DECID 𝑧𝐴 → ∃𝑓 ∈ (2o𝑚 𝑋)(∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = ∅)))
303, 29syl5bi 151 . 2 (𝜑 → (∀𝑥𝑋 DECID 𝑥𝐴 → ∃𝑓 ∈ (2o𝑚 𝑋)(∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = ∅)))
31 omex 4570 . . . . . . . . 9 ω ∈ V
32 2ssom 13684 . . . . . . . . 9 2o ⊆ ω
33 mapss 6657 . . . . . . . . 9 ((ω ∈ V ∧ 2o ⊆ ω) → (2o𝑚 𝑋) ⊆ (ω ↑𝑚 𝑋))
3431, 32, 33mp2an 423 . . . . . . . 8 (2o𝑚 𝑋) ⊆ (ω ↑𝑚 𝑋)
35 fveq1 5485 . . . . . . . . . . . . 13 (𝑓 = 𝑔 → (𝑓𝑥) = (𝑔𝑥))
3635eqeq1d 2174 . . . . . . . . . . . 12 (𝑓 = 𝑔 → ((𝑓𝑥) = 1o ↔ (𝑔𝑥) = 1o))
3736ralbidv 2466 . . . . . . . . . . 11 (𝑓 = 𝑔 → (∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = 1o ↔ ∀𝑥 ∈ (𝑋𝐴)(𝑔𝑥) = 1o))
3835eqeq1d 2174 . . . . . . . . . . . 12 (𝑓 = 𝑔 → ((𝑓𝑥) = ∅ ↔ (𝑔𝑥) = ∅))
3938ralbidv 2466 . . . . . . . . . . 11 (𝑓 = 𝑔 → (∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = ∅ ↔ ∀𝑥 ∈ (𝑋𝐴)(𝑔𝑥) = ∅))
4037, 39anbi12d 465 . . . . . . . . . 10 (𝑓 = 𝑔 → ((∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = ∅) ↔ (∀𝑥 ∈ (𝑋𝐴)(𝑔𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋𝐴)(𝑔𝑥) = ∅)))
4140cbvrexvw 2697 . . . . . . . . 9 (∃𝑓 ∈ (2o𝑚 𝑋)(∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = ∅) ↔ ∃𝑔 ∈ (2o𝑚 𝑋)(∀𝑥 ∈ (𝑋𝐴)(𝑔𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋𝐴)(𝑔𝑥) = ∅))
42 fveqeq2 5495 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → ((𝑔𝑥) = 1o ↔ (𝑔𝑦) = 1o))
4342cbvralvw 2696 . . . . . . . . . . . 12 (∀𝑥 ∈ (𝑋𝐴)(𝑔𝑥) = 1o ↔ ∀𝑦 ∈ (𝑋𝐴)(𝑔𝑦) = 1o)
44 1n0 6400 . . . . . . . . . . . . . . . 16 1o ≠ ∅
4544neii 2338 . . . . . . . . . . . . . . 15 ¬ 1o = ∅
46 eqeq1 2172 . . . . . . . . . . . . . . 15 ((𝑔𝑦) = 1o → ((𝑔𝑦) = ∅ ↔ 1o = ∅))
4745, 46mtbiri 665 . . . . . . . . . . . . . 14 ((𝑔𝑦) = 1o → ¬ (𝑔𝑦) = ∅)
4847neqned 2343 . . . . . . . . . . . . 13 ((𝑔𝑦) = 1o → (𝑔𝑦) ≠ ∅)
4948ralimi 2529 . . . . . . . . . . . 12 (∀𝑦 ∈ (𝑋𝐴)(𝑔𝑦) = 1o → ∀𝑦 ∈ (𝑋𝐴)(𝑔𝑦) ≠ ∅)
5043, 49sylbi 120 . . . . . . . . . . 11 (∀𝑥 ∈ (𝑋𝐴)(𝑔𝑥) = 1o → ∀𝑦 ∈ (𝑋𝐴)(𝑔𝑦) ≠ ∅)
51 fveqeq2 5495 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → ((𝑔𝑥) = ∅ ↔ (𝑔𝑦) = ∅))
5251cbvralvw 2696 . . . . . . . . . . . 12 (∀𝑥 ∈ (𝑋𝐴)(𝑔𝑥) = ∅ ↔ ∀𝑦 ∈ (𝑋𝐴)(𝑔𝑦) = ∅)
5352biimpi 119 . . . . . . . . . . 11 (∀𝑥 ∈ (𝑋𝐴)(𝑔𝑥) = ∅ → ∀𝑦 ∈ (𝑋𝐴)(𝑔𝑦) = ∅)
5450, 53anim12i 336 . . . . . . . . . 10 ((∀𝑥 ∈ (𝑋𝐴)(𝑔𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋𝐴)(𝑔𝑥) = ∅) → (∀𝑦 ∈ (𝑋𝐴)(𝑔𝑦) ≠ ∅ ∧ ∀𝑦 ∈ (𝑋𝐴)(𝑔𝑦) = ∅))
5554reximi 2563 . . . . . . . . 9 (∃𝑔 ∈ (2o𝑚 𝑋)(∀𝑥 ∈ (𝑋𝐴)(𝑔𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋𝐴)(𝑔𝑥) = ∅) → ∃𝑔 ∈ (2o𝑚 𝑋)(∀𝑦 ∈ (𝑋𝐴)(𝑔𝑦) ≠ ∅ ∧ ∀𝑦 ∈ (𝑋𝐴)(𝑔𝑦) = ∅))
5641, 55sylbi 120 . . . . . . . 8 (∃𝑓 ∈ (2o𝑚 𝑋)(∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = ∅) → ∃𝑔 ∈ (2o𝑚 𝑋)(∀𝑦 ∈ (𝑋𝐴)(𝑔𝑦) ≠ ∅ ∧ ∀𝑦 ∈ (𝑋𝐴)(𝑔𝑦) = ∅))
57 ssrexv 3207 . . . . . . . 8 ((2o𝑚 𝑋) ⊆ (ω ↑𝑚 𝑋) → (∃𝑔 ∈ (2o𝑚 𝑋)(∀𝑦 ∈ (𝑋𝐴)(𝑔𝑦) ≠ ∅ ∧ ∀𝑦 ∈ (𝑋𝐴)(𝑔𝑦) = ∅) → ∃𝑔 ∈ (ω ↑𝑚 𝑋)(∀𝑦 ∈ (𝑋𝐴)(𝑔𝑦) ≠ ∅ ∧ ∀𝑦 ∈ (𝑋𝐴)(𝑔𝑦) = ∅)))
5834, 56, 57mpsyl 65 . . . . . . 7 (∃𝑓 ∈ (2o𝑚 𝑋)(∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = ∅) → ∃𝑔 ∈ (ω ↑𝑚 𝑋)(∀𝑦 ∈ (𝑋𝐴)(𝑔𝑦) ≠ ∅ ∧ ∀𝑦 ∈ (𝑋𝐴)(𝑔𝑦) = ∅))
5958adantl 275 . . . . . 6 ((𝜑 ∧ ∃𝑓 ∈ (2o𝑚 𝑋)(∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = ∅)) → ∃𝑔 ∈ (ω ↑𝑚 𝑋)(∀𝑦 ∈ (𝑋𝐴)(𝑔𝑦) ≠ ∅ ∧ ∀𝑦 ∈ (𝑋𝐴)(𝑔𝑦) = ∅))
6059bj-charfunr 13692 . . . . 5 ((𝜑 ∧ ∃𝑓 ∈ (2o𝑚 𝑋)(∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = ∅)) → ∀𝑦𝑋 DECID ¬ 𝑦𝐴)
6160ex 114 . . . 4 (𝜑 → (∃𝑓 ∈ (2o𝑚 𝑋)(∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = ∅) → ∀𝑦𝑋 DECID ¬ 𝑦𝐴))
62 eleq1w 2227 . . . . . . 7 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
6362notbid 657 . . . . . 6 (𝑥 = 𝑦 → (¬ 𝑥𝐴 ↔ ¬ 𝑦𝐴))
6463dcbid 828 . . . . 5 (𝑥 = 𝑦 → (DECID ¬ 𝑥𝐴DECID ¬ 𝑦𝐴))
6564cbvralvw 2696 . . . 4 (∀𝑥𝑋 DECID ¬ 𝑥𝐴 ↔ ∀𝑦𝑋 DECID ¬ 𝑦𝐴)
6661, 65syl6ibr 161 . . 3 (𝜑 → (∃𝑓 ∈ (2o𝑚 𝑋)(∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = ∅) → ∀𝑥𝑋 DECID ¬ 𝑥𝐴))
67 bj-charfunbi.st . . . . . 6 (𝜑 → ∀𝑥𝑋 STAB 𝑥𝐴)
6867r19.21bi 2554 . . . . 5 ((𝜑𝑥𝑋) → STAB 𝑥𝐴)
69 stdcn 837 . . . . 5 (STAB 𝑥𝐴 ↔ (DECID ¬ 𝑥𝐴DECID 𝑥𝐴))
7068, 69sylib 121 . . . 4 ((𝜑𝑥𝑋) → (DECID ¬ 𝑥𝐴DECID 𝑥𝐴))
7170ralimdva 2533 . . 3 (𝜑 → (∀𝑥𝑋 DECID ¬ 𝑥𝐴 → ∀𝑥𝑋 DECID 𝑥𝐴))
7266, 71syld 45 . 2 (𝜑 → (∃𝑓 ∈ (2o𝑚 𝑋)(∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = ∅) → ∀𝑥𝑋 DECID 𝑥𝐴))
7330, 72impbid 128 1 (𝜑 → (∀𝑥𝑋 DECID 𝑥𝐴 ↔ ∃𝑓 ∈ (2o𝑚 𝑋)(∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = ∅)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  STAB wstab 820  DECID wdc 824   = wceq 1343  wcel 2136  wne 2336  wral 2444  wrex 2445  Vcvv 2726  cdif 3113  cin 3115  wss 3116  c0 3409  ifcif 3520  cmpt 4043  Oncon0 4341  ωcom 4567  wf 5184  cfv 5188  (class class class)co 5842  1oc1o 6377  2oc2o 6378  𝑚 cmap 6614
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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565
This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1o 6384  df-2o 6385  df-map 6616
This theorem is referenced by: (None)
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