Theorem ackbij1lem9 8994

Description: Lemma for ackbij1 9004. (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		inss2 3812	. . . . . . . . . 10 ⊢ (𝒫 ω ∩ Fin) ⊆ Fin
2	1	sseli 3579	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → 𝐴 ∈ Fin)
3	2	3ad2ant1 1080	. . . . . . . 8 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐴 ∈ Fin)
4		snfi 7982	. . . . . . . . . 10 ⊢ {𝑦} ∈ Fin
5		inss1 3811	. . . . . . . . . . . . . . . 16 ⊢ (𝒫 ω ∩ Fin) ⊆ 𝒫 ω
6	5	sseli 3579	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → 𝐴 ∈ 𝒫 ω)
7	6	elpwid 4141	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → 𝐴 ⊆ ω)
8	7	3ad2ant1 1080	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐴 ⊆ ω)
9		onfin2 8096	. . . . . . . . . . . . . 14 ⊢ ω = (On ∩ Fin)
10		inss2 3812	. . . . . . . . . . . . . 14 ⊢ (On ∩ Fin) ⊆ Fin
11	9, 10	eqsstri 3614	. . . . . . . . . . . . 13 ⊢ ω ⊆ Fin
12	8, 11	syl6ss 3595	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐴 ⊆ Fin)
13	12	sselda 3583	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ Fin)
14		pwfi 8205	. . . . . . . . . . 11 ⊢ (𝑦 ∈ Fin ↔ 𝒫 𝑦 ∈ Fin)
15	13, 14	sylib 208	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ 𝐴) → 𝒫 𝑦 ∈ Fin)
16		xpfi 8175	. . . . . . . . . 10 ⊢ (({𝑦} ∈ Fin ∧ 𝒫 𝑦 ∈ Fin) → ({𝑦} × 𝒫 𝑦) ∈ Fin)
17	4, 15, 16	sylancr 694	. . . . . . . . 9 ⊢ (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ 𝐴) → ({𝑦} × 𝒫 𝑦) ∈ Fin)
18	17	ralrimiva 2960	. . . . . . . 8 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → ∀𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ∈ Fin)
19		iunfi 8198	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ∈ Fin) → ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ∈ Fin)
20	3, 18, 19	syl2anc 692	. . . . . . 7 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ∈ Fin)
21		ficardid 8732	. . . . . . 7 ⊢ (∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ∈ Fin → (card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) ≈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦))
22	20, 21	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → (card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) ≈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦))
23	1	sseli 3579	. . . . . . . . 9 ⊢ (𝐵 ∈ (𝒫 ω ∩ Fin) → 𝐵 ∈ Fin)
24	23	3ad2ant2 1081	. . . . . . . 8 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐵 ∈ Fin)
25	5	sseli 3579	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ (𝒫 ω ∩ Fin) → 𝐵 ∈ 𝒫 ω)
26	25	elpwid 4141	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ (𝒫 ω ∩ Fin) → 𝐵 ⊆ ω)
27	26	3ad2ant2 1081	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐵 ⊆ ω)
28	27, 11	syl6ss 3595	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐵 ⊆ Fin)
29	28	sselda 3583	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ Fin)
30	29, 14	sylib 208	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ 𝐵) → 𝒫 𝑦 ∈ Fin)
31	4, 30, 16	sylancr 694	. . . . . . . . 9 ⊢ (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ 𝐵) → ({𝑦} × 𝒫 𝑦) ∈ Fin)
32	31	ralrimiva 2960	. . . . . . . 8 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → ∀𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦) ∈ Fin)
33		iunfi 8198	. . . . . . . 8 ⊢ ((𝐵 ∈ Fin ∧ ∀𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦) ∈ Fin) → ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦) ∈ Fin)
34	24, 32, 33	syl2anc 692	. . . . . . 7 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦) ∈ Fin)
35		ficardid 8732	. . . . . . 7 ⊢ (∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦) ∈ Fin → (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)) ≈ ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦))
36	34, 35	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)) ≈ ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦))
37		cdaen 8939	. . . . . 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)) +𝑐 (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦))) ≈ (∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) +𝑐 ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)))
38	22, 36, 37	syl2anc 692	. . . . 5 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → ((card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) +𝑐 (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦))) ≈ (∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) +𝑐 ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)))
39		djudisj 5520	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ∩ ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)) = ∅)
40	39	3ad2ant3 1082	. . . . . . 7 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → (∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ∩ ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)) = ∅)
41		cdaun 8938	. . . . . . 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) +𝑐 ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)) ≈ (∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ∪ ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)))
42	20, 34, 40, 41	syl3anc 1323	. . . . . 6 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → (∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) +𝑐 ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)) ≈ (∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ∪ ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)))
43		iunxun 4571	. . . . . 6 ⊢ ∪ 𝑦 ∈ (𝐴 ∪ 𝐵)({𝑦} × 𝒫 𝑦) = (∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ∪ ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦))
44	42, 43	syl6breqr 4655	. . . . 5 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → (∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) +𝑐 ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)) ≈ ∪ 𝑦 ∈ (𝐴 ∪ 𝐵)({𝑦} × 𝒫 𝑦))
45		entr 7952	. . . . 5 ⊢ ((((card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) +𝑐 (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦))) ≈ (∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) +𝑐 ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)) ∧ (∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) +𝑐 ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)) ≈ ∪ 𝑦 ∈ (𝐴 ∪ 𝐵)({𝑦} × 𝒫 𝑦)) → ((card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) +𝑐 (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦))) ≈ ∪ 𝑦 ∈ (𝐴 ∪ 𝐵)({𝑦} × 𝒫 𝑦))
46	38, 44, 45	syl2anc 692	. . . 4 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → ((card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) +𝑐 (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦))) ≈ ∪ 𝑦 ∈ (𝐴 ∪ 𝐵)({𝑦} × 𝒫 𝑦))
47		carden2b 8737	. . . 4 ⊢ (((card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) +𝑐 (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦))) ≈ ∪ 𝑦 ∈ (𝐴 ∪ 𝐵)({𝑦} × 𝒫 𝑦) → (card‘((card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) +𝑐 (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)))) = (card‘∪ 𝑦 ∈ (𝐴 ∪ 𝐵)({𝑦} × 𝒫 𝑦)))
48	46, 47	syl 17	. . 3 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → (card‘((card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) +𝑐 (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)))) = (card‘∪ 𝑦 ∈ (𝐴 ∪ 𝐵)({𝑦} × 𝒫 𝑦)))
49		ficardom 8731	. . . . 5 ⊢ (∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ∈ Fin → (card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) ∈ ω)
50	20, 49	syl 17	. . . 4 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → (card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) ∈ ω)
51		ficardom 8731	. . . . 5 ⊢ (∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦) ∈ Fin → (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)) ∈ ω)
52	34, 51	syl 17	. . . 4 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)) ∈ ω)
53		nnacda 8967	. . . 4 ⊢ (((card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) ∈ ω ∧ (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)) ∈ ω) → (card‘((card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) +𝑐 (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)))) = ((card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) +𝑜 (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦))))
54	50, 52, 53	syl2anc 692	. . 3 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → (card‘((card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) +𝑐 (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)))) = ((card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) +𝑜 (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦))))
55	48, 54	eqtr3d 2657	. 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))))
56		ackbij1lem6 8991	. . . 4 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → (𝐴 ∪ 𝐵) ∈ (𝒫 ω ∩ Fin))
57	56	3adant3 1079	. . 3 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∪ 𝐵) ∈ (𝒫 ω ∩ Fin))
58		ackbij.f	. . . 4 ⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)))
59	58	ackbij1lem7 8992	. . 3 ⊢ ((𝐴 ∪ 𝐵) ∈ (𝒫 ω ∩ Fin) → (𝐹‘(𝐴 ∪ 𝐵)) = (card‘∪ 𝑦 ∈ (𝐴 ∪ 𝐵)({𝑦} × 𝒫 𝑦)))
60	57, 59	syl 17	. 2 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐹‘(𝐴 ∪ 𝐵)) = (card‘∪ 𝑦 ∈ (𝐴 ∪ 𝐵)({𝑦} × 𝒫 𝑦)))
61	58	ackbij1lem7 8992	. . . 4 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹‘𝐴) = (card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)))
62	58	ackbij1lem7 8992	. . . 4 ⊢ (𝐵 ∈ (𝒫 ω ∩ Fin) → (𝐹‘𝐵) = (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)))
63	61, 62	oveqan12d 6623	. . 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))))
64	63	3adant3 1079	. 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))))
65	55, 60, 64	3eqtr4d 2665	1 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐹‘(𝐴 ∪ 𝐵)) = ((𝐹‘𝐴) +𝑜 (𝐹‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987  ∀wral 2907   ∪ cun 3553   ∩ cin 3554   ⊆ wss 3555  ∅c0 3891  𝒫 cpw 4130  {csn 4148  ∪ ciun 4485   class class class wbr 4613   ↦ cmpt 4673   × cxp 5072  Oncon0 5682  ‘cfv 5847  (class class class)co 6604  ωcom 7012   +_𝑜 coa 7502   ≈ cen 7896  Fincfn 7899  cardccrd 8705   +_𝑐 ccda 8933

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-card 8709  df-cda 8934

This theorem is referenced by:  ackbij1lem12  8997  ackbij1lem13  8998  ackbij1lem14  8999  ackbij1lem16  9001  ackbij1lem18  9003