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Theorem isf34lem7 10374

Description: Lemma for isfin3-4 10377. (Contributed by Stefan O'Rear, 7-Nov-2014.)

**Hypothesis**
Ref	Expression
compss.a	⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥))

**Assertion**
Ref	Expression
isf34lem7	⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦)) → ∪ ran 𝐺 ∈ ran 𝐺)

Distinct variable groups: 𝑥,𝑦,𝐴 𝑦,𝐹 𝑦,𝐺 Allowed substitution hints: 𝐹(𝑥) 𝐺(𝑥)

**Proof of Theorem isf34lem7**
Step	Hyp	Ref	Expression
1		compss.a	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥))
2	1	isf34lem2 10368	. . . . . 6 ⊢ (𝐴 ∈ Fin^III → 𝐹:𝒫 𝐴⟶𝒫 𝐴)
3	2	adantr 482	. . . . 5 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → 𝐹:𝒫 𝐴⟶𝒫 𝐴)
4	3	3adant3 1133	. . . 4 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦)) → 𝐹:𝒫 𝐴⟶𝒫 𝐴)
5	4	ffnd 6719	. . 3 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦)) → 𝐹 Fn 𝒫 𝐴)
6		imassrn 6071	. . . 4 ⊢ (𝐹 “ ran 𝐺) ⊆ ran 𝐹
7	3	frnd 6726	. . . . 5 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → ran 𝐹 ⊆ 𝒫 𝐴)
8	7	3adant3 1133	. . . 4 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦)) → ran 𝐹 ⊆ 𝒫 𝐴)
9	6, 8	sstrid 3994	. . 3 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦)) → (𝐹 “ ran 𝐺) ⊆ 𝒫 𝐴)
10		simp1 1137	. . . . 5 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦)) → 𝐴 ∈ Fin^III)
11		fco 6742	. . . . . . 7 ⊢ ((𝐹:𝒫 𝐴⟶𝒫 𝐴 ∧ 𝐺:ω⟶𝒫 𝐴) → (𝐹 ∘ 𝐺):ω⟶𝒫 𝐴)
12	2, 11	sylan 581	. . . . . 6 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → (𝐹 ∘ 𝐺):ω⟶𝒫 𝐴)
13	12	3adant3 1133	. . . . 5 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦)) → (𝐹 ∘ 𝐺):ω⟶𝒫 𝐴)
14		sscon 4139	. . . . . . . 8 ⊢ ((𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦) → (𝐴 ∖ (𝐺‘suc 𝑦)) ⊆ (𝐴 ∖ (𝐺‘𝑦)))
15		simpr 486	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → 𝐺:ω⟶𝒫 𝐴)
16		peano2 7881	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ω → suc 𝑦 ∈ ω)
17		fvco3 6991	. . . . . . . . . . 11 ⊢ ((𝐺:ω⟶𝒫 𝐴 ∧ suc 𝑦 ∈ ω) → ((𝐹 ∘ 𝐺)‘suc 𝑦) = (𝐹‘(𝐺‘suc 𝑦)))
18	15, 16, 17	syl2an 597	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → ((𝐹 ∘ 𝐺)‘suc 𝑦) = (𝐹‘(𝐺‘suc 𝑦)))
19		simpll 766	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → 𝐴 ∈ Fin^III)
20		ffvelcdm 7084	. . . . . . . . . . . . 13 ⊢ ((𝐺:ω⟶𝒫 𝐴 ∧ suc 𝑦 ∈ ω) → (𝐺‘suc 𝑦) ∈ 𝒫 𝐴)
21	15, 16, 20	syl2an 597	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → (𝐺‘suc 𝑦) ∈ 𝒫 𝐴)
22	21	elpwid 4612	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → (𝐺‘suc 𝑦) ⊆ 𝐴)
23	1	isf34lem1 10367	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin^III ∧ (𝐺‘suc 𝑦) ⊆ 𝐴) → (𝐹‘(𝐺‘suc 𝑦)) = (𝐴 ∖ (𝐺‘suc 𝑦)))
24	19, 22, 23	syl2anc 585	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → (𝐹‘(𝐺‘suc 𝑦)) = (𝐴 ∖ (𝐺‘suc 𝑦)))
25	18, 24	eqtrd 2773	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → ((𝐹 ∘ 𝐺)‘suc 𝑦) = (𝐴 ∖ (𝐺‘suc 𝑦)))
26		fvco3 6991	. . . . . . . . . . 11 ⊢ ((𝐺:ω⟶𝒫 𝐴 ∧ 𝑦 ∈ ω) → ((𝐹 ∘ 𝐺)‘𝑦) = (𝐹‘(𝐺‘𝑦)))
27	26	adantll 713	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → ((𝐹 ∘ 𝐺)‘𝑦) = (𝐹‘(𝐺‘𝑦)))
28		ffvelcdm 7084	. . . . . . . . . . . . 13 ⊢ ((𝐺:ω⟶𝒫 𝐴 ∧ 𝑦 ∈ ω) → (𝐺‘𝑦) ∈ 𝒫 𝐴)
29	28	adantll 713	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → (𝐺‘𝑦) ∈ 𝒫 𝐴)
30	29	elpwid 4612	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → (𝐺‘𝑦) ⊆ 𝐴)
31	1	isf34lem1 10367	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin^III ∧ (𝐺‘𝑦) ⊆ 𝐴) → (𝐹‘(𝐺‘𝑦)) = (𝐴 ∖ (𝐺‘𝑦)))
32	19, 30, 31	syl2anc 585	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → (𝐹‘(𝐺‘𝑦)) = (𝐴 ∖ (𝐺‘𝑦)))
33	27, 32	eqtrd 2773	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → ((𝐹 ∘ 𝐺)‘𝑦) = (𝐴 ∖ (𝐺‘𝑦)))
34	25, 33	sseq12d 4016	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → (((𝐹 ∘ 𝐺)‘suc 𝑦) ⊆ ((𝐹 ∘ 𝐺)‘𝑦) ↔ (𝐴 ∖ (𝐺‘suc 𝑦)) ⊆ (𝐴 ∖ (𝐺‘𝑦))))
35	14, 34	imbitrrid 245	. . . . . . 7 ⊢ (((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → ((𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦) → ((𝐹 ∘ 𝐺)‘suc 𝑦) ⊆ ((𝐹 ∘ 𝐺)‘𝑦)))
36	35	ralimdva 3168	. . . . . 6 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → (∀𝑦 ∈ ω (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦) → ∀𝑦 ∈ ω ((𝐹 ∘ 𝐺)‘suc 𝑦) ⊆ ((𝐹 ∘ 𝐺)‘𝑦)))
37	36	3impia 1118	. . . . 5 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦)) → ∀𝑦 ∈ ω ((𝐹 ∘ 𝐺)‘suc 𝑦) ⊆ ((𝐹 ∘ 𝐺)‘𝑦))
38		fin33i 10364	. . . . 5 ⊢ ((𝐴 ∈ Fin^III ∧ (𝐹 ∘ 𝐺):ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω ((𝐹 ∘ 𝐺)‘suc 𝑦) ⊆ ((𝐹 ∘ 𝐺)‘𝑦)) → ∩ ran (𝐹 ∘ 𝐺) ∈ ran (𝐹 ∘ 𝐺))
39	10, 13, 37, 38	syl3anc 1372	. . . 4 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦)) → ∩ ran (𝐹 ∘ 𝐺) ∈ ran (𝐹 ∘ 𝐺))
40		rnco2 6253	. . . . 5 ⊢ ran (𝐹 ∘ 𝐺) = (𝐹 “ ran 𝐺)
41	40	inteqi 4955	. . . 4 ⊢ ∩ ran (𝐹 ∘ 𝐺) = ∩ (𝐹 “ ran 𝐺)
42	39, 41, 40	3eltr3g 2850	. . 3 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦)) → ∩ (𝐹 “ ran 𝐺) ∈ (𝐹 “ ran 𝐺))
43		fnfvima 7235	. . 3 ⊢ ((𝐹 Fn 𝒫 𝐴 ∧ (𝐹 “ ran 𝐺) ⊆ 𝒫 𝐴 ∧ ∩ (𝐹 “ ran 𝐺) ∈ (𝐹 “ ran 𝐺)) → (𝐹‘∩ (𝐹 “ ran 𝐺)) ∈ (𝐹 “ (𝐹 “ ran 𝐺)))
44	5, 9, 42, 43	syl3anc 1372	. 2 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦)) → (𝐹‘∩ (𝐹 “ ran 𝐺)) ∈ (𝐹 “ (𝐹 “ ran 𝐺)))
45		simpl 484	. . . . . 6 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → 𝐴 ∈ Fin^III)
46	6, 7	sstrid 3994	. . . . . 6 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → (𝐹 “ ran 𝐺) ⊆ 𝒫 𝐴)
47		incom 4202	. . . . . . . . 9 ⊢ (dom 𝐹 ∩ ran 𝐺) = (ran 𝐺 ∩ dom 𝐹)
48		frn 6725	. . . . . . . . . . . 12 ⊢ (𝐺:ω⟶𝒫 𝐴 → ran 𝐺 ⊆ 𝒫 𝐴)
49	48	adantl 483	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → ran 𝐺 ⊆ 𝒫 𝐴)
50	3	fdmd 6729	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → dom 𝐹 = 𝒫 𝐴)
51	49, 50	sseqtrrd 4024	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → ran 𝐺 ⊆ dom 𝐹)
52		df-ss 3966	. . . . . . . . . 10 ⊢ (ran 𝐺 ⊆ dom 𝐹 ↔ (ran 𝐺 ∩ dom 𝐹) = ran 𝐺)
53	51, 52	sylib 217	. . . . . . . . 9 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → (ran 𝐺 ∩ dom 𝐹) = ran 𝐺)
54	47, 53	eqtrid 2785	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → (dom 𝐹 ∩ ran 𝐺) = ran 𝐺)
55		fdm 6727	. . . . . . . . . . 11 ⊢ (𝐺:ω⟶𝒫 𝐴 → dom 𝐺 = ω)
56	55	adantl 483	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → dom 𝐺 = ω)
57		peano1 7879	. . . . . . . . . . 11 ⊢ ∅ ∈ ω
58		ne0i 4335	. . . . . . . . . . 11 ⊢ (∅ ∈ ω → ω ≠ ∅)
59	57, 58	mp1i 13	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → ω ≠ ∅)
60	56, 59	eqnetrd 3009	. . . . . . . . 9 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → dom 𝐺 ≠ ∅)
61		dm0rn0 5925	. . . . . . . . . 10 ⊢ (dom 𝐺 = ∅ ↔ ran 𝐺 = ∅)
62	61	necon3bii 2994	. . . . . . . . 9 ⊢ (dom 𝐺 ≠ ∅ ↔ ran 𝐺 ≠ ∅)
63	60, 62	sylib 217	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → ran 𝐺 ≠ ∅)
64	54, 63	eqnetrd 3009	. . . . . . 7 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → (dom 𝐹 ∩ ran 𝐺) ≠ ∅)
65		imadisj 6080	. . . . . . . 8 ⊢ ((𝐹 “ ran 𝐺) = ∅ ↔ (dom 𝐹 ∩ ran 𝐺) = ∅)
66	65	necon3bii 2994	. . . . . . 7 ⊢ ((𝐹 “ ran 𝐺) ≠ ∅ ↔ (dom 𝐹 ∩ ran 𝐺) ≠ ∅)
67	64, 66	sylibr 233	. . . . . 6 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → (𝐹 “ ran 𝐺) ≠ ∅)
68	1	isf34lem5 10373	. . . . . 6 ⊢ ((𝐴 ∈ Fin^III ∧ ((𝐹 “ ran 𝐺) ⊆ 𝒫 𝐴 ∧ (𝐹 “ ran 𝐺) ≠ ∅)) → (𝐹‘∩ (𝐹 “ ran 𝐺)) = ∪ (𝐹 “ (𝐹 “ ran 𝐺)))
69	45, 46, 67, 68	syl12anc 836	. . . . 5 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → (𝐹‘∩ (𝐹 “ ran 𝐺)) = ∪ (𝐹 “ (𝐹 “ ran 𝐺)))
70	1	isf34lem3 10370	. . . . . . 7 ⊢ ((𝐴 ∈ Fin^III ∧ ran 𝐺 ⊆ 𝒫 𝐴) → (𝐹 “ (𝐹 “ ran 𝐺)) = ran 𝐺)
71	45, 49, 70	syl2anc 585	. . . . . 6 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → (𝐹 “ (𝐹 “ ran 𝐺)) = ran 𝐺)
72	71	unieqd 4923	. . . . 5 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → ∪ (𝐹 “ (𝐹 “ ran 𝐺)) = ∪ ran 𝐺)
73	69, 72	eqtrd 2773	. . . 4 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → (𝐹‘∩ (𝐹 “ ran 𝐺)) = ∪ ran 𝐺)
74	73, 71	eleq12d 2828	. . 3 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → ((𝐹‘∩ (𝐹 “ ran 𝐺)) ∈ (𝐹 “ (𝐹 “ ran 𝐺)) ↔ ∪ ran 𝐺 ∈ ran 𝐺))
75	74	3adant3 1133	. 2 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦)) → ((𝐹‘∩ (𝐹 “ ran 𝐺)) ∈ (𝐹 “ (𝐹 “ ran 𝐺)) ↔ ∪ ran 𝐺 ∈ ran 𝐺))
76	44, 75	mpbid 231	1 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦)) → ∪ ran 𝐺 ∈ ran 𝐺)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 ∖ cdif 3946 ∩ cin 3948 ⊆ wss 3949 ∅c0 4323 𝒫 cpw 4603 ∪ cuni 4909 ∩ cint 4951 ↦ cmpt 5232 dom cdm 5677 ran crn 5678 “ cima 5680 ∘ ccom 5681 suc csuc 6367 Fn wfn 6539 ⟶wf 6540 ‘cfv 6544 ωcom 7855 Fin^IIIcfin3 10276

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-rpss 7713 df-om 7856 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-wdom 9560 df-card 9934 df-fin4 10282 df-fin3 10283

This theorem is referenced by: isf34lem6 10375 fin34i 10376

W3C validator