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Theorem bj-charfunbi 14219
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 2238 . . . . 5 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
21dcbid 838 . . . 4 (𝑥 = 𝑧 → (DECID 𝑥𝐴DECID 𝑧𝐴))
32cbvralvw 2707 . . 3 (∀𝑥𝑋 DECID 𝑥𝐴 ↔ ∀𝑧𝑋 DECID 𝑧𝐴)
4 eleq1w 2238 . . . . . . . . . . . 12 (𝑧 = 𝑥 → (𝑧𝐴𝑥𝐴))
54ifbid 3555 . . . . . . . . . . 11 (𝑧 = 𝑥 → if(𝑧𝐴, 1o, ∅) = if(𝑥𝐴, 1o, ∅))
65cbvmptv 4096 . . . . . . . . . 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 133 . . . . . . . . . 10 (∀𝑧𝑋 DECID 𝑧𝐴 → ∀𝑥𝑋 DECID 𝑥𝐴)
98adantl 277 . . . . . . . . 9 ((𝜑 ∧ ∀𝑧𝑋 DECID 𝑧𝐴) → ∀𝑥𝑋 DECID 𝑥𝐴)
107, 9bj-charfundc 14216 . . . . . . . 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 115 . . . . . . 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 6420 . . . . . . . . . . 11 2o ∈ On
1312a1i 9 . . . . . . . . . 10 (𝜑 → 2o ∈ On)
14 bj-charfunbi.ex . . . . . . . . . 10 (𝜑𝑋𝑉)
1513, 14elmapd 6656 . . . . . . . . 9 (𝜑 → ((𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅)) ∈ (2o𝑚 𝑋) ↔ (𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅)):𝑋⟶2o))
1615biimprd 158 . . . . . . . 8 (𝜑 → ((𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅)):𝑋⟶2o → (𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅)) ∈ (2o𝑚 𝑋)))
1716adantrd 279 . . . . . . 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 124 . . . . 5 ((𝜑 ∧ ∀𝑧𝑋 DECID 𝑧𝐴) → (𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅)) ∈ (2o𝑚 𝑋))
20 fveq1 5510 . . . . . . . . 9 (𝑓 = (𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅)) → (𝑓𝑥) = ((𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅))‘𝑥))
2120eqeq1d 2186 . . . . . . . 8 (𝑓 = (𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅)) → ((𝑓𝑥) = 1o ↔ ((𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅))‘𝑥) = 1o))
2221ralbidv 2477 . . . . . . 7 (𝑓 = (𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅)) → (∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = 1o ↔ ∀𝑥 ∈ (𝑋𝐴)((𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅))‘𝑥) = 1o))
2320eqeq1d 2186 . . . . . . . 8 (𝑓 = (𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅)) → ((𝑓𝑥) = ∅ ↔ ((𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅))‘𝑥) = ∅))
2423ralbidv 2477 . . . . . . 7 (𝑓 = (𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅)) → (∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = ∅ ↔ ∀𝑥 ∈ (𝑋𝐴)((𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅))‘𝑥) = ∅))
2522, 24anbi12d 473 . . . . . 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 277 . . . . 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 114 . . . . 5 ((𝜑 ∧ ∀𝑧𝑋 DECID 𝑧𝐴) → (∀𝑥 ∈ (𝑋𝐴)((𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅))‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋𝐴)((𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅))‘𝑥) = ∅))
2819, 26, 27rspcedvd 2847 . . . 4 ((𝜑 ∧ ∀𝑧𝑋 DECID 𝑧𝐴) → ∃𝑓 ∈ (2o𝑚 𝑋)(∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = ∅))
2928ex 115 . . 3 (𝜑 → (∀𝑧𝑋 DECID 𝑧𝐴 → ∃𝑓 ∈ (2o𝑚 𝑋)(∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = ∅)))
303, 29biimtrid 152 . 2 (𝜑 → (∀𝑥𝑋 DECID 𝑥𝐴 → ∃𝑓 ∈ (2o𝑚 𝑋)(∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = ∅)))
31 omex 4589 . . . . . . . . 9 ω ∈ V
32 2ssom 6519 . . . . . . . . 9 2o ⊆ ω
33 mapss 6685 . . . . . . . . 9 ((ω ∈ V ∧ 2o ⊆ ω) → (2o𝑚 𝑋) ⊆ (ω ↑𝑚 𝑋))
3431, 32, 33mp2an 426 . . . . . . . 8 (2o𝑚 𝑋) ⊆ (ω ↑𝑚 𝑋)
35 fveq1 5510 . . . . . . . . . . . . 13 (𝑓 = 𝑔 → (𝑓𝑥) = (𝑔𝑥))
3635eqeq1d 2186 . . . . . . . . . . . 12 (𝑓 = 𝑔 → ((𝑓𝑥) = 1o ↔ (𝑔𝑥) = 1o))
3736ralbidv 2477 . . . . . . . . . . 11 (𝑓 = 𝑔 → (∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = 1o ↔ ∀𝑥 ∈ (𝑋𝐴)(𝑔𝑥) = 1o))
3835eqeq1d 2186 . . . . . . . . . . . 12 (𝑓 = 𝑔 → ((𝑓𝑥) = ∅ ↔ (𝑔𝑥) = ∅))
3938ralbidv 2477 . . . . . . . . . . 11 (𝑓 = 𝑔 → (∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = ∅ ↔ ∀𝑥 ∈ (𝑋𝐴)(𝑔𝑥) = ∅))
4037, 39anbi12d 473 . . . . . . . . . 10 (𝑓 = 𝑔 → ((∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = ∅) ↔ (∀𝑥 ∈ (𝑋𝐴)(𝑔𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋𝐴)(𝑔𝑥) = ∅)))
4140cbvrexvw 2708 . . . . . . . . 9 (∃𝑓 ∈ (2o𝑚 𝑋)(∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = ∅) ↔ ∃𝑔 ∈ (2o𝑚 𝑋)(∀𝑥 ∈ (𝑋𝐴)(𝑔𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋𝐴)(𝑔𝑥) = ∅))
42 fveqeq2 5520 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → ((𝑔𝑥) = 1o ↔ (𝑔𝑦) = 1o))
4342cbvralvw 2707 . . . . . . . . . . . 12 (∀𝑥 ∈ (𝑋𝐴)(𝑔𝑥) = 1o ↔ ∀𝑦 ∈ (𝑋𝐴)(𝑔𝑦) = 1o)
44 1n0 6427 . . . . . . . . . . . . . . . 16 1o ≠ ∅
4544neii 2349 . . . . . . . . . . . . . . 15 ¬ 1o = ∅
46 eqeq1 2184 . . . . . . . . . . . . . . 15 ((𝑔𝑦) = 1o → ((𝑔𝑦) = ∅ ↔ 1o = ∅))
4745, 46mtbiri 675 . . . . . . . . . . . . . 14 ((𝑔𝑦) = 1o → ¬ (𝑔𝑦) = ∅)
4847neqned 2354 . . . . . . . . . . . . 13 ((𝑔𝑦) = 1o → (𝑔𝑦) ≠ ∅)
4948ralimi 2540 . . . . . . . . . . . 12 (∀𝑦 ∈ (𝑋𝐴)(𝑔𝑦) = 1o → ∀𝑦 ∈ (𝑋𝐴)(𝑔𝑦) ≠ ∅)
5043, 49sylbi 121 . . . . . . . . . . 11 (∀𝑥 ∈ (𝑋𝐴)(𝑔𝑥) = 1o → ∀𝑦 ∈ (𝑋𝐴)(𝑔𝑦) ≠ ∅)
51 fveqeq2 5520 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → ((𝑔𝑥) = ∅ ↔ (𝑔𝑦) = ∅))
5251cbvralvw 2707 . . . . . . . . . . . 12 (∀𝑥 ∈ (𝑋𝐴)(𝑔𝑥) = ∅ ↔ ∀𝑦 ∈ (𝑋𝐴)(𝑔𝑦) = ∅)
5352biimpi 120 . . . . . . . . . . 11 (∀𝑥 ∈ (𝑋𝐴)(𝑔𝑥) = ∅ → ∀𝑦 ∈ (𝑋𝐴)(𝑔𝑦) = ∅)
5450, 53anim12i 338 . . . . . . . . . 10 ((∀𝑥 ∈ (𝑋𝐴)(𝑔𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋𝐴)(𝑔𝑥) = ∅) → (∀𝑦 ∈ (𝑋𝐴)(𝑔𝑦) ≠ ∅ ∧ ∀𝑦 ∈ (𝑋𝐴)(𝑔𝑦) = ∅))
5554reximi 2574 . . . . . . . . 9 (∃𝑔 ∈ (2o𝑚 𝑋)(∀𝑥 ∈ (𝑋𝐴)(𝑔𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋𝐴)(𝑔𝑥) = ∅) → ∃𝑔 ∈ (2o𝑚 𝑋)(∀𝑦 ∈ (𝑋𝐴)(𝑔𝑦) ≠ ∅ ∧ ∀𝑦 ∈ (𝑋𝐴)(𝑔𝑦) = ∅))
5641, 55sylbi 121 . . . . . . . 8 (∃𝑓 ∈ (2o𝑚 𝑋)(∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = ∅) → ∃𝑔 ∈ (2o𝑚 𝑋)(∀𝑦 ∈ (𝑋𝐴)(𝑔𝑦) ≠ ∅ ∧ ∀𝑦 ∈ (𝑋𝐴)(𝑔𝑦) = ∅))
57 ssrexv 3220 . . . . . . . 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 277 . . . . . 6 ((𝜑 ∧ ∃𝑓 ∈ (2o𝑚 𝑋)(∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = ∅)) → ∃𝑔 ∈ (ω ↑𝑚 𝑋)(∀𝑦 ∈ (𝑋𝐴)(𝑔𝑦) ≠ ∅ ∧ ∀𝑦 ∈ (𝑋𝐴)(𝑔𝑦) = ∅))
6059bj-charfunr 14218 . . . . 5 ((𝜑 ∧ ∃𝑓 ∈ (2o𝑚 𝑋)(∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = ∅)) → ∀𝑦𝑋 DECID ¬ 𝑦𝐴)
6160ex 115 . . . 4 (𝜑 → (∃𝑓 ∈ (2o𝑚 𝑋)(∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = ∅) → ∀𝑦𝑋 DECID ¬ 𝑦𝐴))
62 eleq1w 2238 . . . . . . 7 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
6362notbid 667 . . . . . 6 (𝑥 = 𝑦 → (¬ 𝑥𝐴 ↔ ¬ 𝑦𝐴))
6463dcbid 838 . . . . 5 (𝑥 = 𝑦 → (DECID ¬ 𝑥𝐴DECID ¬ 𝑦𝐴))
6564cbvralvw 2707 . . . 4 (∀𝑥𝑋 DECID ¬ 𝑥𝐴 ↔ ∀𝑦𝑋 DECID ¬ 𝑦𝐴)
6661, 65syl6ibr 162 . . 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 2565 . . . . 5 ((𝜑𝑥𝑋) → STAB 𝑥𝐴)
69 stdcn 847 . . . . 5 (STAB 𝑥𝐴 ↔ (DECID ¬ 𝑥𝐴DECID 𝑥𝐴))
7068, 69sylib 122 . . . 4 ((𝜑𝑥𝑋) → (DECID ¬ 𝑥𝐴DECID 𝑥𝐴))
7170ralimdva 2544 . . 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 129 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 104  wb 105  STAB wstab 830  DECID wdc 834   = wceq 1353  wcel 2148  wne 2347  wral 2455  wrex 2456  Vcvv 2737  cdif 3126  cin 3128  wss 3129  c0 3422  ifcif 3534  cmpt 4061  Oncon0 4360  ωcom 4586  wf 5208  cfv 5212  (class class class)co 5869  1oc1o 6404  2oc2o 6405  𝑚 cmap 6642
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 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584
This theorem depends on definitions:  df-bi 117  df-stab 831  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-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1o 6411  df-2o 6412  df-map 6644
This theorem is referenced by: (None)
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