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Theorem ackbij1lem9 10259
Description: Lemma for ackbij1 10269. (Contributed by Stefan O'Rear, 19-Nov-2014.)
Hypothesis
Ref Expression
ackbij.f 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))
Assertion
Ref Expression
ackbij1lem9 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → (𝐹‘(𝐴𝐵)) = ((𝐹𝐴) +o (𝐹𝐵)))
Distinct variable groups:   𝑥,𝐹,𝑦   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦

Proof of Theorem ackbij1lem9
StepHypRef Expression
1 elinel2 4194 . . . . . . . . 9 (𝐴 ∈ (𝒫 ω ∩ Fin) → 𝐴 ∈ Fin)
213ad2ant1 1130 . . . . . . . 8 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → 𝐴 ∈ Fin)
3 snfi 9070 . . . . . . . . . 10 {𝑦} ∈ Fin
4 elinel1 4193 . . . . . . . . . . . . . . 15 (𝐴 ∈ (𝒫 ω ∩ Fin) → 𝐴 ∈ 𝒫 ω)
54elpwid 4606 . . . . . . . . . . . . . 14 (𝐴 ∈ (𝒫 ω ∩ Fin) → 𝐴 ⊆ ω)
653ad2ant1 1130 . . . . . . . . . . . . 13 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → 𝐴 ⊆ ω)
7 onfin2 9255 . . . . . . . . . . . . . 14 ω = (On ∩ Fin)
8 inss2 4228 . . . . . . . . . . . . . 14 (On ∩ Fin) ⊆ Fin
97, 8eqsstri 4013 . . . . . . . . . . . . 13 ω ⊆ Fin
106, 9sstrdi 3991 . . . . . . . . . . . 12 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → 𝐴 ⊆ Fin)
1110sselda 3978 . . . . . . . . . . 11 (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) ∧ 𝑦𝐴) → 𝑦 ∈ Fin)
12 pwfi 9349 . . . . . . . . . . 11 (𝑦 ∈ Fin ↔ 𝒫 𝑦 ∈ Fin)
1311, 12sylib 217 . . . . . . . . . 10 (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) ∧ 𝑦𝐴) → 𝒫 𝑦 ∈ Fin)
14 xpfi 9350 . . . . . . . . . 10 (({𝑦} ∈ Fin ∧ 𝒫 𝑦 ∈ Fin) → ({𝑦} × 𝒫 𝑦) ∈ Fin)
153, 13, 14sylancr 585 . . . . . . . . 9 (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) ∧ 𝑦𝐴) → ({𝑦} × 𝒫 𝑦) ∈ Fin)
1615ralrimiva 3136 . . . . . . . 8 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → ∀𝑦𝐴 ({𝑦} × 𝒫 𝑦) ∈ Fin)
17 iunfi 9375 . . . . . . . 8 ((𝐴 ∈ Fin ∧ ∀𝑦𝐴 ({𝑦} × 𝒫 𝑦) ∈ Fin) → 𝑦𝐴 ({𝑦} × 𝒫 𝑦) ∈ Fin)
182, 16, 17syl2anc 582 . . . . . . 7 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → 𝑦𝐴 ({𝑦} × 𝒫 𝑦) ∈ Fin)
19 ficardid 9995 . . . . . . 7 ( 𝑦𝐴 ({𝑦} × 𝒫 𝑦) ∈ Fin → (card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)) ≈ 𝑦𝐴 ({𝑦} × 𝒫 𝑦))
2018, 19syl 17 . . . . . 6 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → (card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)) ≈ 𝑦𝐴 ({𝑦} × 𝒫 𝑦))
21 elinel2 4194 . . . . . . . . 9 (𝐵 ∈ (𝒫 ω ∩ Fin) → 𝐵 ∈ Fin)
22213ad2ant2 1131 . . . . . . . 8 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → 𝐵 ∈ Fin)
23 elinel1 4193 . . . . . . . . . . . . . . 15 (𝐵 ∈ (𝒫 ω ∩ Fin) → 𝐵 ∈ 𝒫 ω)
2423elpwid 4606 . . . . . . . . . . . . . 14 (𝐵 ∈ (𝒫 ω ∩ Fin) → 𝐵 ⊆ ω)
25243ad2ant2 1131 . . . . . . . . . . . . 13 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → 𝐵 ⊆ ω)
2625, 9sstrdi 3991 . . . . . . . . . . . 12 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → 𝐵 ⊆ Fin)
2726sselda 3978 . . . . . . . . . . 11 (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) ∧ 𝑦𝐵) → 𝑦 ∈ Fin)
2827, 12sylib 217 . . . . . . . . . 10 (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) ∧ 𝑦𝐵) → 𝒫 𝑦 ∈ Fin)
293, 28, 14sylancr 585 . . . . . . . . 9 (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) ∧ 𝑦𝐵) → ({𝑦} × 𝒫 𝑦) ∈ Fin)
3029ralrimiva 3136 . . . . . . . 8 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → ∀𝑦𝐵 ({𝑦} × 𝒫 𝑦) ∈ Fin)
31 iunfi 9375 . . . . . . . 8 ((𝐵 ∈ Fin ∧ ∀𝑦𝐵 ({𝑦} × 𝒫 𝑦) ∈ Fin) → 𝑦𝐵 ({𝑦} × 𝒫 𝑦) ∈ Fin)
3222, 30, 31syl2anc 582 . . . . . . 7 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → 𝑦𝐵 ({𝑦} × 𝒫 𝑦) ∈ Fin)
33 ficardid 9995 . . . . . . 7 ( 𝑦𝐵 ({𝑦} × 𝒫 𝑦) ∈ Fin → (card‘ 𝑦𝐵 ({𝑦} × 𝒫 𝑦)) ≈ 𝑦𝐵 ({𝑦} × 𝒫 𝑦))
3432, 33syl 17 . . . . . 6 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → (card‘ 𝑦𝐵 ({𝑦} × 𝒫 𝑦)) ≈ 𝑦𝐵 ({𝑦} × 𝒫 𝑦))
35 djuen 10202 . . . . . 6 (((card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)) ≈ 𝑦𝐴 ({𝑦} × 𝒫 𝑦) ∧ (card‘ 𝑦𝐵 ({𝑦} × 𝒫 𝑦)) ≈ 𝑦𝐵 ({𝑦} × 𝒫 𝑦)) → ((card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)) ⊔ (card‘ 𝑦𝐵 ({𝑦} × 𝒫 𝑦))) ≈ ( 𝑦𝐴 ({𝑦} × 𝒫 𝑦) ⊔ 𝑦𝐵 ({𝑦} × 𝒫 𝑦)))
3620, 34, 35syl2anc 582 . . . . 5 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → ((card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)) ⊔ (card‘ 𝑦𝐵 ({𝑦} × 𝒫 𝑦))) ≈ ( 𝑦𝐴 ({𝑦} × 𝒫 𝑦) ⊔ 𝑦𝐵 ({𝑦} × 𝒫 𝑦)))
37 djudisj 6168 . . . . . . . 8 ((𝐴𝐵) = ∅ → ( 𝑦𝐴 ({𝑦} × 𝒫 𝑦) ∩ 𝑦𝐵 ({𝑦} × 𝒫 𝑦)) = ∅)
38373ad2ant3 1132 . . . . . . 7 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → ( 𝑦𝐴 ({𝑦} × 𝒫 𝑦) ∩ 𝑦𝐵 ({𝑦} × 𝒫 𝑦)) = ∅)
39 endjudisj 10201 . . . . . . 7 (( 𝑦𝐴 ({𝑦} × 𝒫 𝑦) ∈ Fin ∧ 𝑦𝐵 ({𝑦} × 𝒫 𝑦) ∈ Fin ∧ ( 𝑦𝐴 ({𝑦} × 𝒫 𝑦) ∩ 𝑦𝐵 ({𝑦} × 𝒫 𝑦)) = ∅) → ( 𝑦𝐴 ({𝑦} × 𝒫 𝑦) ⊔ 𝑦𝐵 ({𝑦} × 𝒫 𝑦)) ≈ ( 𝑦𝐴 ({𝑦} × 𝒫 𝑦) ∪ 𝑦𝐵 ({𝑦} × 𝒫 𝑦)))
4018, 32, 38, 39syl3anc 1368 . . . . . 6 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → ( 𝑦𝐴 ({𝑦} × 𝒫 𝑦) ⊔ 𝑦𝐵 ({𝑦} × 𝒫 𝑦)) ≈ ( 𝑦𝐴 ({𝑦} × 𝒫 𝑦) ∪ 𝑦𝐵 ({𝑦} × 𝒫 𝑦)))
41 iunxun 5094 . . . . . 6 𝑦 ∈ (𝐴𝐵)({𝑦} × 𝒫 𝑦) = ( 𝑦𝐴 ({𝑦} × 𝒫 𝑦) ∪ 𝑦𝐵 ({𝑦} × 𝒫 𝑦))
4240, 41breqtrrdi 5185 . . . . 5 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → ( 𝑦𝐴 ({𝑦} × 𝒫 𝑦) ⊔ 𝑦𝐵 ({𝑦} × 𝒫 𝑦)) ≈ 𝑦 ∈ (𝐴𝐵)({𝑦} × 𝒫 𝑦))
43 entr 9026 . . . . 5 ((((card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)) ⊔ (card‘ 𝑦𝐵 ({𝑦} × 𝒫 𝑦))) ≈ ( 𝑦𝐴 ({𝑦} × 𝒫 𝑦) ⊔ 𝑦𝐵 ({𝑦} × 𝒫 𝑦)) ∧ ( 𝑦𝐴 ({𝑦} × 𝒫 𝑦) ⊔ 𝑦𝐵 ({𝑦} × 𝒫 𝑦)) ≈ 𝑦 ∈ (𝐴𝐵)({𝑦} × 𝒫 𝑦)) → ((card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)) ⊔ (card‘ 𝑦𝐵 ({𝑦} × 𝒫 𝑦))) ≈ 𝑦 ∈ (𝐴𝐵)({𝑦} × 𝒫 𝑦))
4436, 42, 43syl2anc 582 . . . 4 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → ((card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)) ⊔ (card‘ 𝑦𝐵 ({𝑦} × 𝒫 𝑦))) ≈ 𝑦 ∈ (𝐴𝐵)({𝑦} × 𝒫 𝑦))
45 carden2b 10000 . . . 4 (((card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)) ⊔ (card‘ 𝑦𝐵 ({𝑦} × 𝒫 𝑦))) ≈ 𝑦 ∈ (𝐴𝐵)({𝑦} × 𝒫 𝑦) → (card‘((card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)) ⊔ (card‘ 𝑦𝐵 ({𝑦} × 𝒫 𝑦)))) = (card‘ 𝑦 ∈ (𝐴𝐵)({𝑦} × 𝒫 𝑦)))
4644, 45syl 17 . . 3 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → (card‘((card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)) ⊔ (card‘ 𝑦𝐵 ({𝑦} × 𝒫 𝑦)))) = (card‘ 𝑦 ∈ (𝐴𝐵)({𝑦} × 𝒫 𝑦)))
47 ficardom 9994 . . . . 5 ( 𝑦𝐴 ({𝑦} × 𝒫 𝑦) ∈ Fin → (card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)) ∈ ω)
4818, 47syl 17 . . . 4 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → (card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)) ∈ ω)
49 ficardom 9994 . . . . 5 ( 𝑦𝐵 ({𝑦} × 𝒫 𝑦) ∈ Fin → (card‘ 𝑦𝐵 ({𝑦} × 𝒫 𝑦)) ∈ ω)
5032, 49syl 17 . . . 4 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → (card‘ 𝑦𝐵 ({𝑦} × 𝒫 𝑦)) ∈ ω)
51 nnadju 10230 . . . 4 (((card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)) ∈ ω ∧ (card‘ 𝑦𝐵 ({𝑦} × 𝒫 𝑦)) ∈ ω) → (card‘((card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)) ⊔ (card‘ 𝑦𝐵 ({𝑦} × 𝒫 𝑦)))) = ((card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)) +o (card‘ 𝑦𝐵 ({𝑦} × 𝒫 𝑦))))
5248, 50, 51syl2anc 582 . . 3 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → (card‘((card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)) ⊔ (card‘ 𝑦𝐵 ({𝑦} × 𝒫 𝑦)))) = ((card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)) +o (card‘ 𝑦𝐵 ({𝑦} × 𝒫 𝑦))))
5346, 52eqtr3d 2768 . 2 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → (card‘ 𝑦 ∈ (𝐴𝐵)({𝑦} × 𝒫 𝑦)) = ((card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)) +o (card‘ 𝑦𝐵 ({𝑦} × 𝒫 𝑦))))
54 ackbij1lem6 10256 . . . 4 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → (𝐴𝐵) ∈ (𝒫 ω ∩ Fin))
55543adant3 1129 . . 3 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → (𝐴𝐵) ∈ (𝒫 ω ∩ Fin))
56 ackbij.f . . . 4 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))
5756ackbij1lem7 10257 . . 3 ((𝐴𝐵) ∈ (𝒫 ω ∩ Fin) → (𝐹‘(𝐴𝐵)) = (card‘ 𝑦 ∈ (𝐴𝐵)({𝑦} × 𝒫 𝑦)))
5855, 57syl 17 . 2 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → (𝐹‘(𝐴𝐵)) = (card‘ 𝑦 ∈ (𝐴𝐵)({𝑦} × 𝒫 𝑦)))
5956ackbij1lem7 10257 . . . 4 (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹𝐴) = (card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)))
6056ackbij1lem7 10257 . . . 4 (𝐵 ∈ (𝒫 ω ∩ Fin) → (𝐹𝐵) = (card‘ 𝑦𝐵 ({𝑦} × 𝒫 𝑦)))
6159, 60oveqan12d 7432 . . 3 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → ((𝐹𝐴) +o (𝐹𝐵)) = ((card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)) +o (card‘ 𝑦𝐵 ({𝑦} × 𝒫 𝑦))))
62613adant3 1129 . 2 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → ((𝐹𝐴) +o (𝐹𝐵)) = ((card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)) +o (card‘ 𝑦𝐵 ({𝑦} × 𝒫 𝑦))))
6353, 58, 623eqtr4d 2776 1 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → (𝐹‘(𝐴𝐵)) = ((𝐹𝐴) +o (𝐹𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084   = wceq 1534  wcel 2099  wral 3051  cun 3944  cin 3945  wss 3946  c0 4322  𝒫 cpw 4597  {csn 4623   ciun 4993   class class class wbr 5143  cmpt 5226   × cxp 5670  Oncon0 6365  cfv 6543  (class class class)co 7413  ωcom 7865   +o coa 8482  cen 8960  Fincfn 8963  cdju 9931  cardccrd 9968
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7735
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-int 4947  df-iun 4995  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6302  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7866  df-1st 7992  df-2nd 7993  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-oadd 8489  df-er 8723  df-en 8964  df-dom 8965  df-sdom 8966  df-fin 8967  df-dju 9934  df-card 9972
This theorem is referenced by:  ackbij1lem12  10262  ackbij1lem13  10263  ackbij1lem14  10264  ackbij1lem16  10266  ackbij1lem18  10268
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