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Theorem ackbij1lem9 10164
Description: Lemma for ackbij1 10174. (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 4156 . . . . . . . . 9 (𝐴 ∈ (𝒫 ω ∩ Fin) → 𝐴 ∈ Fin)
213ad2ant1 1133 . . . . . . . 8 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → 𝐴 ∈ Fin)
3 snfi 8988 . . . . . . . . . 10 {𝑦} ∈ Fin
4 elinel1 4155 . . . . . . . . . . . . . . 15 (𝐴 ∈ (𝒫 ω ∩ Fin) → 𝐴 ∈ 𝒫 ω)
54elpwid 4569 . . . . . . . . . . . . . 14 (𝐴 ∈ (𝒫 ω ∩ Fin) → 𝐴 ⊆ ω)
653ad2ant1 1133 . . . . . . . . . . . . 13 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → 𝐴 ⊆ ω)
7 onfin2 9175 . . . . . . . . . . . . . 14 ω = (On ∩ Fin)
8 inss2 4189 . . . . . . . . . . . . . 14 (On ∩ Fin) ⊆ Fin
97, 8eqsstri 3978 . . . . . . . . . . . . 13 ω ⊆ Fin
106, 9sstrdi 3956 . . . . . . . . . . . 12 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → 𝐴 ⊆ Fin)
1110sselda 3944 . . . . . . . . . . 11 (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) ∧ 𝑦𝐴) → 𝑦 ∈ Fin)
12 pwfi 9122 . . . . . . . . . . 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 9261 . . . . . . . . . 10 (({𝑦} ∈ Fin ∧ 𝒫 𝑦 ∈ Fin) → ({𝑦} × 𝒫 𝑦) ∈ Fin)
153, 13, 14sylancr 587 . . . . . . . . 9 (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) ∧ 𝑦𝐴) → ({𝑦} × 𝒫 𝑦) ∈ Fin)
1615ralrimiva 3143 . . . . . . . 8 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → ∀𝑦𝐴 ({𝑦} × 𝒫 𝑦) ∈ Fin)
17 iunfi 9284 . . . . . . . 8 ((𝐴 ∈ Fin ∧ ∀𝑦𝐴 ({𝑦} × 𝒫 𝑦) ∈ Fin) → 𝑦𝐴 ({𝑦} × 𝒫 𝑦) ∈ Fin)
182, 16, 17syl2anc 584 . . . . . . 7 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → 𝑦𝐴 ({𝑦} × 𝒫 𝑦) ∈ Fin)
19 ficardid 9898 . . . . . . 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 4156 . . . . . . . . 9 (𝐵 ∈ (𝒫 ω ∩ Fin) → 𝐵 ∈ Fin)
22213ad2ant2 1134 . . . . . . . 8 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → 𝐵 ∈ Fin)
23 elinel1 4155 . . . . . . . . . . . . . . 15 (𝐵 ∈ (𝒫 ω ∩ Fin) → 𝐵 ∈ 𝒫 ω)
2423elpwid 4569 . . . . . . . . . . . . . 14 (𝐵 ∈ (𝒫 ω ∩ Fin) → 𝐵 ⊆ ω)
25243ad2ant2 1134 . . . . . . . . . . . . 13 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → 𝐵 ⊆ ω)
2625, 9sstrdi 3956 . . . . . . . . . . . 12 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → 𝐵 ⊆ Fin)
2726sselda 3944 . . . . . . . . . . 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 587 . . . . . . . . 9 (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) ∧ 𝑦𝐵) → ({𝑦} × 𝒫 𝑦) ∈ Fin)
3029ralrimiva 3143 . . . . . . . 8 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → ∀𝑦𝐵 ({𝑦} × 𝒫 𝑦) ∈ Fin)
31 iunfi 9284 . . . . . . . 8 ((𝐵 ∈ Fin ∧ ∀𝑦𝐵 ({𝑦} × 𝒫 𝑦) ∈ Fin) → 𝑦𝐵 ({𝑦} × 𝒫 𝑦) ∈ Fin)
3222, 30, 31syl2anc 584 . . . . . . 7 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → 𝑦𝐵 ({𝑦} × 𝒫 𝑦) ∈ Fin)
33 ficardid 9898 . . . . . . 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 10105 . . . . . 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 584 . . . . 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 6119 . . . . . . . 8 ((𝐴𝐵) = ∅ → ( 𝑦𝐴 ({𝑦} × 𝒫 𝑦) ∩ 𝑦𝐵 ({𝑦} × 𝒫 𝑦)) = ∅)
38373ad2ant3 1135 . . . . . . 7 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → ( 𝑦𝐴 ({𝑦} × 𝒫 𝑦) ∩ 𝑦𝐵 ({𝑦} × 𝒫 𝑦)) = ∅)
39 endjudisj 10104 . . . . . . 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 1371 . . . . . 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 5054 . . . . . 6 𝑦 ∈ (𝐴𝐵)({𝑦} × 𝒫 𝑦) = ( 𝑦𝐴 ({𝑦} × 𝒫 𝑦) ∪ 𝑦𝐵 ({𝑦} × 𝒫 𝑦))
4240, 41breqtrrdi 5147 . . . . 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 8946 . . . . 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 584 . . . 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 9903 . . . 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 9897 . . . . 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 9897 . . . . 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 10133 . . . 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 584 . . 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 2778 . 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 10161 . . . 4 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → (𝐴𝐵) ∈ (𝒫 ω ∩ Fin))
55543adant3 1132 . . 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 10162 . . 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 10162 . . . 4 (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹𝐴) = (card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)))
6056ackbij1lem7 10162 . . . 4 (𝐵 ∈ (𝒫 ω ∩ Fin) → (𝐹𝐵) = (card‘ 𝑦𝐵 ({𝑦} × 𝒫 𝑦)))
6159, 60oveqan12d 7376 . . 3 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → ((𝐹𝐴) +o (𝐹𝐵)) = ((card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)) +o (card‘ 𝑦𝐵 ({𝑦} × 𝒫 𝑦))))
62613adant3 1132 . 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 2786 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 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  cun 3908  cin 3909  wss 3910  c0 4282  𝒫 cpw 4560  {csn 4586   ciun 4954   class class class wbr 5105  cmpt 5188   × cxp 5631  Oncon0 6317  cfv 6496  (class class class)co 7357  ωcom 7802   +o coa 8409  cen 8880  Fincfn 8883  cdju 9834  cardccrd 9871
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-oadd 8416  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-dju 9837  df-card 9875
This theorem is referenced by:  ackbij1lem12  10167  ackbij1lem13  10168  ackbij1lem14  10169  ackbij1lem16  10171  ackbij1lem18  10173
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