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‘∪ 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 4089	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → 𝐴 ∈ Fin)
2	1	3ad2ant1 1124	. . . . . . . 8 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐴 ∈ Fin)
3		snfi 8432	. . . . . . . . . 10 ⊢ {𝑦} ∈ Fin
4		elinel1 4088	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → 𝐴 ∈ 𝒫 ω)
5	4	elpwid 4459	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → 𝐴 ⊆ ω)
6	5	3ad2ant1 1124	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐴 ⊆ ω)
7		onfin2 8546	. . . . . . . . . . . . . 14 ⊢ ω = (On ∩ Fin)
8		inss2 4121	. . . . . . . . . . . . . 14 ⊢ (On ∩ Fin) ⊆ Fin
9	7, 8	eqsstri 3917	. . . . . . . . . . . . 13 ⊢ ω ⊆ Fin
10	6, 9	syl6ss 3896	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐴 ⊆ Fin)
11	10	sselda 3884	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ Fin)
12		pwfi 8655	. . . . . . . . . . 11 ⊢ (𝑦 ∈ Fin ↔ 𝒫 𝑦 ∈ Fin)
13	11, 12	sylib 219	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ 𝐴) → 𝒫 𝑦 ∈ Fin)
14		xpfi 8625	. . . . . . . . . 10 ⊢ (({𝑦} ∈ Fin ∧ 𝒫 𝑦 ∈ Fin) → ({𝑦} × 𝒫 𝑦) ∈ Fin)
15	3, 13, 14	sylancr 587	. . . . . . . . 9 ⊢ (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ 𝐴) → ({𝑦} × 𝒫 𝑦) ∈ Fin)
16	15	ralrimiva 3147	. . . . . . . 8 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → ∀𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ∈ Fin)
17		iunfi 8648	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ∈ Fin) → ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ∈ Fin)
18	2, 16, 17	syl2anc 584	. . . . . . 7 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ∈ Fin)
19		ficardid 9226	. . . . . . 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 4089	. . . . . . . . 9 ⊢ (𝐵 ∈ (𝒫 ω ∩ Fin) → 𝐵 ∈ Fin)
22	21	3ad2ant2 1125	. . . . . . . 8 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐵 ∈ Fin)
23		elinel1 4088	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ (𝒫 ω ∩ Fin) → 𝐵 ∈ 𝒫 ω)
24	23	elpwid 4459	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ (𝒫 ω ∩ Fin) → 𝐵 ⊆ ω)
25	24	3ad2ant2 1125	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐵 ⊆ ω)
26	25, 9	syl6ss 3896	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐵 ⊆ Fin)
27	26	sselda 3884	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ Fin)
28	27, 12	sylib 219	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ 𝐵) → 𝒫 𝑦 ∈ Fin)
29	3, 28, 14	sylancr 587	. . . . . . . . 9 ⊢ (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ 𝐵) → ({𝑦} × 𝒫 𝑦) ∈ Fin)
30	29	ralrimiva 3147	. . . . . . . 8 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → ∀𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦) ∈ Fin)
31		iunfi 8648	. . . . . . . 8 ⊢ ((𝐵 ∈ Fin ∧ ∀𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦) ∈ Fin) → ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦) ∈ Fin)
32	22, 30, 31	syl2anc 584	. . . . . . 7 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦) ∈ Fin)
33		ficardid 9226	. . . . . . 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 9430	. . . . . 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 584	. . . . 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 5892	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ∩ ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)) = ∅)
38	37	3ad2ant3 1126	. . . . . . 7 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → (∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ∩ ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)) = ∅)
39		endjudisj 9429	. . . . . . 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 1362	. . . . . 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 4909	. . . . . 6 ⊢ ∪ 𝑦 ∈ (𝐴 ∪ 𝐵)({𝑦} × 𝒫 𝑦) = (∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ∪ ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦))
42	40, 41	syl6breqr 4998	. . . . 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 8399	. . . . 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 584	. . . 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 9231	. . . 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 9225	. . . . 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 9225	. . . . 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 9458	. . . 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 584	. . 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 2831	. 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 9482	. . . 4 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → (𝐴 ∪ 𝐵) ∈ (𝒫 ω ∩ Fin))
55	54	3adant3 1123	. . 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 9483	. . 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 9483	. . . 4 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹‘𝐴) = (card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)))
60	56	ackbij1lem7 9483	. . . 4 ⊢ (𝐵 ∈ (𝒫 ω ∩ Fin) → (𝐹‘𝐵) = (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)))
61	59, 60	oveqan12d 7026	. . 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 1123	. 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 2839	1 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐹‘(𝐴 ∪ 𝐵)) = ((𝐹‘𝐴) +o (𝐹‘𝐵)))

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