Theorem ackbij1lem9 10183

Description: Lemma for ackbij1 10193. (Contributed by Stefan O'Rear, 19-Nov-2014.)

Hypothesis

Ref	Expression
ackbij.f	⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)))

Assertion

Ref	Expression
ackbij1lem9	⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐹‘(𝐴 ∪ 𝐵)) = ((𝐹‘𝐴) +o (𝐹‘𝐵)))

Distinct variable groups:   𝑥,𝐹,𝑦   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦

Proof of Theorem ackbij1lem9

Step	Hyp	Ref	Expression
1		elinel2 4154	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → 𝐴 ∈ Fin)
2	1	3ad2ant1 1146	. . . . . . . 8 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐴 ∈ Fin)
3		snfi 9024	. . . . . . . . . 10 ⊢ {𝑦} ∈ Fin
4		elinel1 4153	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → 𝐴 ∈ 𝒫 ω)
5	4	elpwid 4564	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → 𝐴 ⊆ ω)
6	5	3ad2ant1 1146	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐴 ⊆ ω)
7		onfin2 9185	. . . . . . . . . . . . . 14 ⊢ ω = (On ∩ Fin)
8		inss2 4189	. . . . . . . . . . . . . 14 ⊢ (On ∩ Fin) ⊆ Fin
9	7, 8	eqsstri 3982	. . . . . . . . . . . . 13 ⊢ ω ⊆ Fin
10	6, 9	sstrdi 3948	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐴 ⊆ Fin)
11	10	sselda 3936	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ Fin)
12		pwfi 9263	. . . . . . . . . . 11 ⊢ (𝑦 ∈ Fin ↔ 𝒫 𝑦 ∈ Fin)
13	11, 12	sylib 220	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ 𝐴) → 𝒫 𝑦 ∈ Fin)
14		xpfi 9264	. . . . . . . . . 10 ⊢ (({𝑦} ∈ Fin ∧ 𝒫 𝑦 ∈ Fin) → ({𝑦} × 𝒫 𝑦) ∈ Fin)
15	3, 13, 14	sylancr 596	. . . . . . . . 9 ⊢ (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ 𝐴) → ({𝑦} × 𝒫 𝑦) ∈ Fin)
16	15	ralrimiva 3154	. . . . . . . 8 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → ∀𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ∈ Fin)
17		iunfi 9286	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ∈ Fin) → ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ∈ Fin)
18	2, 16, 17	syl2anc 593	. . . . . . 7 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ∈ Fin)
19		ficardid 9920	. . . . . . 7 ⊢ (∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ∈ Fin → (card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) ≈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦))
20	18, 19	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → (card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) ≈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦))
21		elinel2 4154	. . . . . . . . 9 ⊢ (𝐵 ∈ (𝒫 ω ∩ Fin) → 𝐵 ∈ Fin)
22	21	3ad2ant2 1147	. . . . . . . 8 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐵 ∈ Fin)
23		elinel1 4153	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ (𝒫 ω ∩ Fin) → 𝐵 ∈ 𝒫 ω)
24	23	elpwid 4564	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ (𝒫 ω ∩ Fin) → 𝐵 ⊆ ω)
25	24	3ad2ant2 1147	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐵 ⊆ ω)
26	25, 9	sstrdi 3948	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐵 ⊆ Fin)
27	26	sselda 3936	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ Fin)
28	27, 12	sylib 220	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ 𝐵) → 𝒫 𝑦 ∈ Fin)
29	3, 28, 14	sylancr 596	. . . . . . . . 9 ⊢ (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ 𝐵) → ({𝑦} × 𝒫 𝑦) ∈ Fin)
30	29	ralrimiva 3154	. . . . . . . 8 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → ∀𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦) ∈ Fin)
31		iunfi 9286	. . . . . . . 8 ⊢ ((𝐵 ∈ Fin ∧ ∀𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦) ∈ Fin) → ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦) ∈ Fin)
32	22, 30, 31	syl2anc 593	. . . . . . 7 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦) ∈ Fin)
33		ficardid 9920	. . . . . . 7 ⊢ (∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦) ∈ Fin → (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)) ≈ ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦))
34	32, 33	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)) ≈ ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦))
35		djuen 10126	. . . . . 6 ⊢ (((card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) ≈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ∧ (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)) ≈ ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)) → ((card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) ⊔ (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦))) ≈ (∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ⊔ ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)))
36	20, 34, 35	syl2anc 593	. . . . 5 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → ((card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) ⊔ (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦))) ≈ (∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ⊔ ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)))
37		djudisj 6152	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ∩ ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)) = ∅)
38	37	3ad2ant3 1148	. . . . . . 7 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → (∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ∩ ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)) = ∅)
39		endjudisj 10125	. . . . . . 7 ⊢ ((∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ∈ Fin ∧ ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦) ∈ Fin ∧ (∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ∩ ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)) = ∅) → (∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ⊔ ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)) ≈ (∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ∪ ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)))
40	18, 32, 38, 39	syl3anc 1390	. . . . . 6 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → (∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ⊔ ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)) ≈ (∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ∪ ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)))
41		iunxun 5051	. . . . . 6 ⊢ ∪ 𝑦 ∈ (𝐴 ∪ 𝐵)({𝑦} × 𝒫 𝑦) = (∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ∪ ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦))
42	40, 41	breqtrrdi 5142	. . . . 5 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → (∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ⊔ ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)) ≈ ∪ 𝑦 ∈ (𝐴 ∪ 𝐵)({𝑦} × 𝒫 𝑦))
43		entr 8987	. . . . 5 ⊢ ((((card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) ⊔ (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦))) ≈ (∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ⊔ ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)) ∧ (∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ⊔ ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)) ≈ ∪ 𝑦 ∈ (𝐴 ∪ 𝐵)({𝑦} × 𝒫 𝑦)) → ((card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) ⊔ (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦))) ≈ ∪ 𝑦 ∈ (𝐴 ∪ 𝐵)({𝑦} × 𝒫 𝑦))
44	36, 42, 43	syl2anc 593	. . . 4 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → ((card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) ⊔ (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦))) ≈ ∪ 𝑦 ∈ (𝐴 ∪ 𝐵)({𝑦} × 𝒫 𝑦))
45		carden2b 9925	. . . 4 ⊢ (((card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) ⊔ (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦))) ≈ ∪ 𝑦 ∈ (𝐴 ∪ 𝐵)({𝑦} × 𝒫 𝑦) → (card‘((card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) ⊔ (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)))) = (card‘∪ 𝑦 ∈ (𝐴 ∪ 𝐵)({𝑦} × 𝒫 𝑦)))
46	44, 45	syl 17	. . 3 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → (card‘((card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) ⊔ (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)))) = (card‘∪ 𝑦 ∈ (𝐴 ∪ 𝐵)({𝑦} × 𝒫 𝑦)))
47		ficardom 9919	. . . . 5 ⊢ (∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ∈ Fin → (card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) ∈ ω)
48	18, 47	syl 17	. . . 4 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → (card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) ∈ ω)
49		ficardom 9919	. . . . 5 ⊢ (∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦) ∈ Fin → (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)) ∈ ω)
50	32, 49	syl 17	. . . 4 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)) ∈ ω)
51		nnadju 10154	. . . 4 ⊢ (((card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) ∈ ω ∧ (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)) ∈ ω) → (card‘((card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) ⊔ (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)))) = ((card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) +o (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦))))
52	48, 50, 51	syl2anc 593	. . 3 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → (card‘((card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) ⊔ (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)))) = ((card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) +o (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦))))
53	46, 52	eqtr3d 2799	. 2 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → (card‘∪ 𝑦 ∈ (𝐴 ∪ 𝐵)({𝑦} × 𝒫 𝑦)) = ((card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) +o (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦))))
54		ackbij1lem6 10180	. . . 4 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → (𝐴 ∪ 𝐵) ∈ (𝒫 ω ∩ Fin))
55	54	3adant3 1145	. . 3 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∪ 𝐵) ∈ (𝒫 ω ∩ Fin))
56		ackbij.f	. . . 4 ⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)))
57	56	ackbij1lem7 10181	. . 3 ⊢ ((𝐴 ∪ 𝐵) ∈ (𝒫 ω ∩ Fin) → (𝐹‘(𝐴 ∪ 𝐵)) = (card‘∪ 𝑦 ∈ (𝐴 ∪ 𝐵)({𝑦} × 𝒫 𝑦)))
58	55, 57	syl 17	. 2 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐹‘(𝐴 ∪ 𝐵)) = (card‘∪ 𝑦 ∈ (𝐴 ∪ 𝐵)({𝑦} × 𝒫 𝑦)))
59	56	ackbij1lem7 10181	. . . 4 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹‘𝐴) = (card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)))
60	56	ackbij1lem7 10181	. . . 4 ⊢ (𝐵 ∈ (𝒫 ω ∩ Fin) → (𝐹‘𝐵) = (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)))
61	59, 60	oveqan12d 7415	. . 3 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → ((𝐹‘𝐴) +o (𝐹‘𝐵)) = ((card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) +o (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦))))
62	61	3adant3 1145	. 2 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐹‘𝐴) +o (𝐹‘𝐵)) = ((card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) +o (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦))))
63	53, 58, 62	3eqtr4d 2807	1 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐹‘(𝐴 ∪ 𝐵)) = ((𝐹‘𝐴) +o (𝐹‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1098   = wceq 1560   ∈ wcel 2142  ∀wral 3076   ∪ cun 3902   ∩ cin 3903   ⊆ wss 3904  ∅c0 4285  𝒫 cpw 4555  {csn 4582  ∪ ciun 4949   class class class wbr 5100   ↦ cmpt 5181   × cxp 5645  Oncon0 6346  ‘cfv 6521  (class class class)co 7396  ωcom 7846   +_o coa 8434   ≈ cen 8924  Fincfn 8927   ⊔ cdju 9856  cardccrd 9893

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4906  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-om 7847  df-1st 7970  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-1o 8437  df-oadd 8441  df-er 8678  df-en 8928  df-dom 8929  df-sdom 8930  df-fin 8931  df-dju 9859  df-card 9897

This theorem is referenced by:  ackbij1lem12  10186  ackbij1lem13  10187  ackbij1lem14  10188  ackbij1lem16  10190  ackbij1lem18  10192