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Theorem ackbij1lem9 9485
Description: Lemma for ackbij1 9495. (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 4089 . . . . . . . . 9 (𝐴 ∈ (𝒫 ω ∩ Fin) → 𝐴 ∈ Fin)
213ad2ant1 1124 . . . . . . . 8 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → 𝐴 ∈ Fin)
3 snfi 8432 . . . . . . . . . 10 {𝑦} ∈ Fin
4 elinel1 4088 . . . . . . . . . . . . . . 15 (𝐴 ∈ (𝒫 ω ∩ Fin) → 𝐴 ∈ 𝒫 ω)
54elpwid 4459 . . . . . . . . . . . . . 14 (𝐴 ∈ (𝒫 ω ∩ Fin) → 𝐴 ⊆ ω)
653ad2ant1 1124 . . . . . . . . . . . . 13 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → 𝐴 ⊆ ω)
7 onfin2 8546 . . . . . . . . . . . . . 14 ω = (On ∩ Fin)
8 inss2 4121 . . . . . . . . . . . . . 14 (On ∩ Fin) ⊆ Fin
97, 8eqsstri 3917 . . . . . . . . . . . . 13 ω ⊆ Fin
106, 9syl6ss 3896 . . . . . . . . . . . 12 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → 𝐴 ⊆ Fin)
1110sselda 3884 . . . . . . . . . . 11 (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) ∧ 𝑦𝐴) → 𝑦 ∈ Fin)
12 pwfi 8655 . . . . . . . . . . 11 (𝑦 ∈ Fin ↔ 𝒫 𝑦 ∈ Fin)
1311, 12sylib 219 . . . . . . . . . 10 (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) ∧ 𝑦𝐴) → 𝒫 𝑦 ∈ Fin)
14 xpfi 8625 . . . . . . . . . 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 3147 . . . . . . . 8 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → ∀𝑦𝐴 ({𝑦} × 𝒫 𝑦) ∈ Fin)
17 iunfi 8648 . . . . . . . 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 9226 . . . . . . 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 4089 . . . . . . . . 9 (𝐵 ∈ (𝒫 ω ∩ Fin) → 𝐵 ∈ Fin)
22213ad2ant2 1125 . . . . . . . 8 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → 𝐵 ∈ Fin)
23 elinel1 4088 . . . . . . . . . . . . . . 15 (𝐵 ∈ (𝒫 ω ∩ Fin) → 𝐵 ∈ 𝒫 ω)
2423elpwid 4459 . . . . . . . . . . . . . 14 (𝐵 ∈ (𝒫 ω ∩ Fin) → 𝐵 ⊆ ω)
25243ad2ant2 1125 . . . . . . . . . . . . 13 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → 𝐵 ⊆ ω)
2625, 9syl6ss 3896 . . . . . . . . . . . 12 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → 𝐵 ⊆ Fin)
2726sselda 3884 . . . . . . . . . . 11 (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) ∧ 𝑦𝐵) → 𝑦 ∈ Fin)
2827, 12sylib 219 . . . . . . . . . 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 3147 . . . . . . . 8 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → ∀𝑦𝐵 ({𝑦} × 𝒫 𝑦) ∈ Fin)
31 iunfi 8648 . . . . . . . 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 9226 . . . . . . 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 9430 . . . . . 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 5892 . . . . . . . 8 ((𝐴𝐵) = ∅ → ( 𝑦𝐴 ({𝑦} × 𝒫 𝑦) ∩ 𝑦𝐵 ({𝑦} × 𝒫 𝑦)) = ∅)
38373ad2ant3 1126 . . . . . . 7 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → ( 𝑦𝐴 ({𝑦} × 𝒫 𝑦) ∩ 𝑦𝐵 ({𝑦} × 𝒫 𝑦)) = ∅)
39 endjudisj 9429 . . . . . . 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 1362 . . . . . 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 4909 . . . . . 6 𝑦 ∈ (𝐴𝐵)({𝑦} × 𝒫 𝑦) = ( 𝑦𝐴 ({𝑦} × 𝒫 𝑦) ∪ 𝑦𝐵 ({𝑦} × 𝒫 𝑦))
4240, 41syl6breqr 4998 . . . . 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 8399 . . . . 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 9231 . . . 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 9225 . . . . 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 9225 . . . . 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 9458 . . . 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 2831 . 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 9482 . . . 4 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → (𝐴𝐵) ∈ (𝒫 ω ∩ Fin))
55543adant3 1123 . . 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 9483 . . 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 9483 . . . 4 (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹𝐴) = (card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)))
6056ackbij1lem7 9483 . . . 4 (𝐵 ∈ (𝒫 ω ∩ Fin) → (𝐹𝐵) = (card‘ 𝑦𝐵 ({𝑦} × 𝒫 𝑦)))
6159, 60oveqan12d 7026 . . 3 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → ((𝐹𝐴) +o (𝐹𝐵)) = ((card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)) +o (card‘ 𝑦𝐵 ({𝑦} × 𝒫 𝑦))))
62613adant3 1123 . 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 2839 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 1078   = wceq 1520  wcel 2079  wral 3103  cun 3852  cin 3853  wss 3854  c0 4206  𝒫 cpw 4447  {csn 4466   ciun 4819   class class class wbr 4956  cmpt 5035   × cxp 5433  Oncon0 6058  cfv 6217  (class class class)co 7007  ωcom 7427   +o coa 7941  cen 8344  Fincfn 8347  cdju 9162  cardccrd 9199
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-rep 5075  ax-sep 5088  ax-nul 5095  ax-pow 5150  ax-pr 5214  ax-un 7310
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1079  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ne 2983  df-ral 3108  df-rex 3109  df-reu 3110  df-rmo 3111  df-rab 3112  df-v 3434  df-sbc 3702  df-csb 3807  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-pss 3871  df-nul 4207  df-if 4376  df-pw 4449  df-sn 4467  df-pr 4469  df-tp 4471  df-op 4473  df-uni 4740  df-int 4777  df-iun 4821  df-br 4957  df-opab 5019  df-mpt 5036  df-tr 5058  df-id 5340  df-eprel 5345  df-po 5354  df-so 5355  df-fr 5394  df-we 5396  df-xp 5441  df-rel 5442  df-cnv 5443  df-co 5444  df-dm 5445  df-rn 5446  df-res 5447  df-ima 5448  df-pred 6015  df-ord 6061  df-on 6062  df-lim 6063  df-suc 6064  df-iota 6181  df-fun 6219  df-fn 6220  df-f 6221  df-f1 6222  df-fo 6223  df-f1o 6224  df-fv 6225  df-ov 7010  df-oprab 7011  df-mpo 7012  df-om 7428  df-1st 7536  df-2nd 7537  df-wrecs 7789  df-recs 7851  df-rdg 7889  df-1o 7944  df-2o 7945  df-oadd 7948  df-er 8130  df-map 8249  df-en 8348  df-dom 8349  df-sdom 8350  df-fin 8351  df-dju 9165  df-card 9203
This theorem is referenced by:  ackbij1lem12  9488  ackbij1lem13  9489  ackbij1lem14  9490  ackbij1lem16  9492  ackbij1lem18  9494
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