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Theorem bj-charfunbi 13846
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 2231 . . . . 5 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
21dcbid 833 . . . 4 (𝑥 = 𝑧 → (DECID 𝑥𝐴DECID 𝑧𝐴))
32cbvralvw 2700 . . 3 (∀𝑥𝑋 DECID 𝑥𝐴 ↔ ∀𝑧𝑋 DECID 𝑧𝐴)
4 eleq1w 2231 . . . . . . . . . . . 12 (𝑧 = 𝑥 → (𝑧𝐴𝑥𝐴))
54ifbid 3547 . . . . . . . . . . 11 (𝑧 = 𝑥 → if(𝑧𝐴, 1o, ∅) = if(𝑥𝐴, 1o, ∅))
65cbvmptv 4085 . . . . . . . . . 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 13843 . . . . . . . 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 6404 . . . . . . . . . . 11 2o ∈ On
1312a1i 9 . . . . . . . . . 10 (𝜑 → 2o ∈ On)
14 bj-charfunbi.ex . . . . . . . . . 10 (𝜑𝑋𝑉)
1513, 14elmapd 6640 . . . . . . . . 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 5495 . . . . . . . . 9 (𝑓 = (𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅)) → (𝑓𝑥) = ((𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅))‘𝑥))
2120eqeq1d 2179 . . . . . . . 8 (𝑓 = (𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅)) → ((𝑓𝑥) = 1o ↔ ((𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅))‘𝑥) = 1o))
2221ralbidv 2470 . . . . . . 7 (𝑓 = (𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅)) → (∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = 1o ↔ ∀𝑥 ∈ (𝑋𝐴)((𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅))‘𝑥) = 1o))
2320eqeq1d 2179 . . . . . . . 8 (𝑓 = (𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅)) → ((𝑓𝑥) = ∅ ↔ ((𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅))‘𝑥) = ∅))
2423ralbidv 2470 . . . . . . 7 (𝑓 = (𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅)) → (∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = ∅ ↔ ∀𝑥 ∈ (𝑋𝐴)((𝑧𝑋 ↦ if(𝑧𝐴, 1o, ∅))‘𝑥) = ∅))
2522, 24anbi12d 470 . . . . . 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 2840 . . . 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 4577 . . . . . . . . 9 ω ∈ V
32 2ssom 6503 . . . . . . . . 9 2o ⊆ ω
33 mapss 6669 . . . . . . . . 9 ((ω ∈ V ∧ 2o ⊆ ω) → (2o𝑚 𝑋) ⊆ (ω ↑𝑚 𝑋))
3431, 32, 33mp2an 424 . . . . . . . 8 (2o𝑚 𝑋) ⊆ (ω ↑𝑚 𝑋)
35 fveq1 5495 . . . . . . . . . . . . 13 (𝑓 = 𝑔 → (𝑓𝑥) = (𝑔𝑥))
3635eqeq1d 2179 . . . . . . . . . . . 12 (𝑓 = 𝑔 → ((𝑓𝑥) = 1o ↔ (𝑔𝑥) = 1o))
3736ralbidv 2470 . . . . . . . . . . 11 (𝑓 = 𝑔 → (∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = 1o ↔ ∀𝑥 ∈ (𝑋𝐴)(𝑔𝑥) = 1o))
3835eqeq1d 2179 . . . . . . . . . . . 12 (𝑓 = 𝑔 → ((𝑓𝑥) = ∅ ↔ (𝑔𝑥) = ∅))
3938ralbidv 2470 . . . . . . . . . . 11 (𝑓 = 𝑔 → (∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = ∅ ↔ ∀𝑥 ∈ (𝑋𝐴)(𝑔𝑥) = ∅))
4037, 39anbi12d 470 . . . . . . . . . 10 (𝑓 = 𝑔 → ((∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = ∅) ↔ (∀𝑥 ∈ (𝑋𝐴)(𝑔𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋𝐴)(𝑔𝑥) = ∅)))
4140cbvrexvw 2701 . . . . . . . . 9 (∃𝑓 ∈ (2o𝑚 𝑋)(∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋𝐴)(𝑓𝑥) = ∅) ↔ ∃𝑔 ∈ (2o𝑚 𝑋)(∀𝑥 ∈ (𝑋𝐴)(𝑔𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋𝐴)(𝑔𝑥) = ∅))
42 fveqeq2 5505 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → ((𝑔𝑥) = 1o ↔ (𝑔𝑦) = 1o))
4342cbvralvw 2700 . . . . . . . . . . . 12 (∀𝑥 ∈ (𝑋𝐴)(𝑔𝑥) = 1o ↔ ∀𝑦 ∈ (𝑋𝐴)(𝑔𝑦) = 1o)
44 1n0 6411 . . . . . . . . . . . . . . . 16 1o ≠ ∅
4544neii 2342 . . . . . . . . . . . . . . 15 ¬ 1o = ∅
46 eqeq1 2177 . . . . . . . . . . . . . . 15 ((𝑔𝑦) = 1o → ((𝑔𝑦) = ∅ ↔ 1o = ∅))
4745, 46mtbiri 670 . . . . . . . . . . . . . 14 ((𝑔𝑦) = 1o → ¬ (𝑔𝑦) = ∅)
4847neqned 2347 . . . . . . . . . . . . 13 ((𝑔𝑦) = 1o → (𝑔𝑦) ≠ ∅)
4948ralimi 2533 . . . . . . . . . . . 12 (∀𝑦 ∈ (𝑋𝐴)(𝑔𝑦) = 1o → ∀𝑦 ∈ (𝑋𝐴)(𝑔𝑦) ≠ ∅)
5043, 49sylbi 120 . . . . . . . . . . 11 (∀𝑥 ∈ (𝑋𝐴)(𝑔𝑥) = 1o → ∀𝑦 ∈ (𝑋𝐴)(𝑔𝑦) ≠ ∅)
51 fveqeq2 5505 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → ((𝑔𝑥) = ∅ ↔ (𝑔𝑦) = ∅))
5251cbvralvw 2700 . . . . . . . . . . . 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 2567 . . . . . . . . 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 3212 . . . . . . . 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 13845 . . . . 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 2231 . . . . . . 7 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
6362notbid 662 . . . . . 6 (𝑥 = 𝑦 → (¬ 𝑥𝐴 ↔ ¬ 𝑦𝐴))
6463dcbid 833 . . . . 5 (𝑥 = 𝑦 → (DECID ¬ 𝑥𝐴DECID ¬ 𝑦𝐴))
6564cbvralvw 2700 . . . 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 2558 . . . . 5 ((𝜑𝑥𝑋) → STAB 𝑥𝐴)
69 stdcn 842 . . . . 5 (STAB 𝑥𝐴 ↔ (DECID ¬ 𝑥𝐴DECID 𝑥𝐴))
7068, 69sylib 121 . . . 4 ((𝜑𝑥𝑋) → (DECID ¬ 𝑥𝐴DECID 𝑥𝐴))
7170ralimdva 2537 . . 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 825  DECID wdc 829   = wceq 1348  wcel 2141  wne 2340  wral 2448  wrex 2449  Vcvv 2730  cdif 3118  cin 3120  wss 3121  c0 3414  ifcif 3526  cmpt 4050  Oncon0 4348  ωcom 4574  wf 5194  cfv 5198  (class class class)co 5853  1oc1o 6388  2oc2o 6389  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-stab 826  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-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  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-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  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-res 4623  df-ima 4624  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-1o 6395  df-2o 6396  df-map 6628
This theorem is referenced by: (None)
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