Theorem ackbij1lem9 9842

Description: Lemma for ackbij1 9852. (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 4110	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → 𝐴 ∈ Fin)
2	1	3ad2ant1 1135	. . . . . . . 8 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐴 ∈ Fin)
3		snfi 8721	. . . . . . . . . 10 ⊢ {𝑦} ∈ Fin
4		elinel1 4109	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → 𝐴 ∈ 𝒫 ω)
5	4	elpwid 4524	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → 𝐴 ⊆ ω)
6	5	3ad2ant1 1135	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐴 ⊆ ω)
7		onfin2 8871	. . . . . . . . . . . . . 14 ⊢ ω = (On ∩ Fin)
8		inss2 4144	. . . . . . . . . . . . . 14 ⊢ (On ∩ Fin) ⊆ Fin
9	7, 8	eqsstri 3935	. . . . . . . . . . . . 13 ⊢ ω ⊆ Fin
10	6, 9	sstrdi 3913	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐴 ⊆ Fin)
11	10	sselda 3901	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ Fin)
12		pwfi 8856	. . . . . . . . . . 11 ⊢ (𝑦 ∈ Fin ↔ 𝒫 𝑦 ∈ Fin)
13	11, 12	sylib 221	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ 𝐴) → 𝒫 𝑦 ∈ Fin)
14		xpfi 8942	. . . . . . . . . 10 ⊢ (({𝑦} ∈ Fin ∧ 𝒫 𝑦 ∈ Fin) → ({𝑦} × 𝒫 𝑦) ∈ Fin)
15	3, 13, 14	sylancr 590	. . . . . . . . 9 ⊢ (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ 𝐴) → ({𝑦} × 𝒫 𝑦) ∈ Fin)
16	15	ralrimiva 3105	. . . . . . . 8 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → ∀𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ∈ Fin)
17		iunfi 8964	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ∈ Fin) → ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ∈ Fin)
18	2, 16, 17	syl2anc 587	. . . . . . 7 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ∈ Fin)
19		ficardid 9578	. . . . . . 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 4110	. . . . . . . . 9 ⊢ (𝐵 ∈ (𝒫 ω ∩ Fin) → 𝐵 ∈ Fin)
22	21	3ad2ant2 1136	. . . . . . . 8 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐵 ∈ Fin)
23		elinel1 4109	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ (𝒫 ω ∩ Fin) → 𝐵 ∈ 𝒫 ω)
24	23	elpwid 4524	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ (𝒫 ω ∩ Fin) → 𝐵 ⊆ ω)
25	24	3ad2ant2 1136	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐵 ⊆ ω)
26	25, 9	sstrdi 3913	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐵 ⊆ Fin)
27	26	sselda 3901	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ Fin)
28	27, 12	sylib 221	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ 𝐵) → 𝒫 𝑦 ∈ Fin)
29	3, 28, 14	sylancr 590	. . . . . . . . 9 ⊢ (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ 𝐵) → ({𝑦} × 𝒫 𝑦) ∈ Fin)
30	29	ralrimiva 3105	. . . . . . . 8 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → ∀𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦) ∈ Fin)
31		iunfi 8964	. . . . . . . 8 ⊢ ((𝐵 ∈ Fin ∧ ∀𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦) ∈ Fin) → ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦) ∈ Fin)
32	22, 30, 31	syl2anc 587	. . . . . . 7 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦) ∈ Fin)
33		ficardid 9578	. . . . . . 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 9783	. . . . . 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 587	. . . . 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 6030	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ∩ ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)) = ∅)
38	37	3ad2ant3 1137	. . . . . . 7 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → (∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ∩ ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)) = ∅)
39		endjudisj 9782	. . . . . . 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 1373	. . . . . 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 5002	. . . . . 6 ⊢ ∪ 𝑦 ∈ (𝐴 ∪ 𝐵)({𝑦} × 𝒫 𝑦) = (∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ∪ ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦))
42	40, 41	breqtrrdi 5095	. . . . 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 8680	. . . . 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 587	. . . 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 9583	. . . 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 9577	. . . . 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 9577	. . . . 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 9811	. . . 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 587	. . 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 2779	. 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 9839	. . . 4 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → (𝐴 ∪ 𝐵) ∈ (𝒫 ω ∩ Fin))
55	54	3adant3 1134	. . 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 9840	. . 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 9840	. . . 4 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹‘𝐴) = (card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)))
60	56	ackbij1lem7 9840	. . . 4 ⊢ (𝐵 ∈ (𝒫 ω ∩ Fin) → (𝐹‘𝐵) = (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)))
61	59, 60	oveqan12d 7232	. . 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 1134	. 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 2787	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 1089   = wceq 1543   ∈ wcel 2110  ∀wral 3061   ∪ cun 3864   ∩ cin 3865   ⊆ wss 3866  ∅c0 4237  𝒫 cpw 4513  {csn 4541  ∪ ciun 4904   class class class wbr 5053   ↦ cmpt 5135   × cxp 5549  Oncon0 6213  ‘cfv 6380  (class class class)co 7213  ωcom 7644   +_o coa 8199   ≈ cen 8623  Fincfn 8626   ⊔ cdju 9514  cardccrd 9551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-oadd 8206  df-er 8391  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-dju 9517  df-card 9555

This theorem is referenced by:  ackbij1lem12  9845  ackbij1lem13  9846  ackbij1lem14  9847  ackbij1lem16  9849  ackbij1lem18  9851