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Theorem isf34lem7 10291

Description: Lemma for isfin3-4 10294. (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 10285	. . . . . 6 ⊢ (𝐴 ∈ Fin^III → 𝐹:𝒫 𝐴⟶𝒫 𝐴)
3	2	adantr 480	. . . . 5 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → 𝐹:𝒫 𝐴⟶𝒫 𝐴)
4	3	3adant3 1132	. . . 4 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦)) → 𝐹:𝒫 𝐴⟶𝒫 𝐴)
5	4	ffnd 6663	. . 3 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦)) → 𝐹 Fn 𝒫 𝐴)
6		imassrn 6030	. . . 4 ⊢ (𝐹 “ ran 𝐺) ⊆ ran 𝐹
7	3	frnd 6670	. . . . 5 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → ran 𝐹 ⊆ 𝒫 𝐴)
8	7	3adant3 1132	. . . 4 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦)) → ran 𝐹 ⊆ 𝒫 𝐴)
9	6, 8	sstrid 3945	. . 3 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦)) → (𝐹 “ ran 𝐺) ⊆ 𝒫 𝐴)
10		simp1 1136	. . . . 5 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦)) → 𝐴 ∈ Fin^III)
11		fco 6686	. . . . . . 7 ⊢ ((𝐹:𝒫 𝐴⟶𝒫 𝐴 ∧ 𝐺:ω⟶𝒫 𝐴) → (𝐹 ∘ 𝐺):ω⟶𝒫 𝐴)
12	2, 11	sylan 580	. . . . . 6 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → (𝐹 ∘ 𝐺):ω⟶𝒫 𝐴)
13	12	3adant3 1132	. . . . 5 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦)) → (𝐹 ∘ 𝐺):ω⟶𝒫 𝐴)
14		sscon 4095	. . . . . . . 8 ⊢ ((𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦) → (𝐴 ∖ (𝐺‘suc 𝑦)) ⊆ (𝐴 ∖ (𝐺‘𝑦)))
15		simpr 484	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → 𝐺:ω⟶𝒫 𝐴)
16		peano2 7832	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ω → suc 𝑦 ∈ ω)
17		fvco3 6933	. . . . . . . . . . 11 ⊢ ((𝐺:ω⟶𝒫 𝐴 ∧ suc 𝑦 ∈ ω) → ((𝐹 ∘ 𝐺)‘suc 𝑦) = (𝐹‘(𝐺‘suc 𝑦)))
18	15, 16, 17	syl2an 596	. . . . . . . . . 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 7026	. . . . . . . . . . . . 13 ⊢ ((𝐺:ω⟶𝒫 𝐴 ∧ suc 𝑦 ∈ ω) → (𝐺‘suc 𝑦) ∈ 𝒫 𝐴)
21	15, 16, 20	syl2an 596	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → (𝐺‘suc 𝑦) ∈ 𝒫 𝐴)
22	21	elpwid 4563	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → (𝐺‘suc 𝑦) ⊆ 𝐴)
23	1	isf34lem1 10284	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin^III ∧ (𝐺‘suc 𝑦) ⊆ 𝐴) → (𝐹‘(𝐺‘suc 𝑦)) = (𝐴 ∖ (𝐺‘suc 𝑦)))
24	19, 22, 23	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → (𝐹‘(𝐺‘suc 𝑦)) = (𝐴 ∖ (𝐺‘suc 𝑦)))
25	18, 24	eqtrd 2771	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → ((𝐹 ∘ 𝐺)‘suc 𝑦) = (𝐴 ∖ (𝐺‘suc 𝑦)))
26		fvco3 6933	. . . . . . . . . . 11 ⊢ ((𝐺:ω⟶𝒫 𝐴 ∧ 𝑦 ∈ ω) → ((𝐹 ∘ 𝐺)‘𝑦) = (𝐹‘(𝐺‘𝑦)))
27	26	adantll 714	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → ((𝐹 ∘ 𝐺)‘𝑦) = (𝐹‘(𝐺‘𝑦)))
28		ffvelcdm 7026	. . . . . . . . . . . . 13 ⊢ ((𝐺:ω⟶𝒫 𝐴 ∧ 𝑦 ∈ ω) → (𝐺‘𝑦) ∈ 𝒫 𝐴)
29	28	adantll 714	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → (𝐺‘𝑦) ∈ 𝒫 𝐴)
30	29	elpwid 4563	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → (𝐺‘𝑦) ⊆ 𝐴)
31	1	isf34lem1 10284	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin^III ∧ (𝐺‘𝑦) ⊆ 𝐴) → (𝐹‘(𝐺‘𝑦)) = (𝐴 ∖ (𝐺‘𝑦)))
32	19, 30, 31	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → (𝐹‘(𝐺‘𝑦)) = (𝐴 ∖ (𝐺‘𝑦)))
33	27, 32	eqtrd 2771	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → ((𝐹 ∘ 𝐺)‘𝑦) = (𝐴 ∖ (𝐺‘𝑦)))
34	25, 33	sseq12d 3967	. . . . . . . 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 246	. . . . . . 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 3148	. . . . . 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 1117	. . . . 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 10281	. . . . 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 1373	. . . 4 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦)) → ∩ ran (𝐹 ∘ 𝐺) ∈ ran (𝐹 ∘ 𝐺))
40		rnco2 6212	. . . . 5 ⊢ ran (𝐹 ∘ 𝐺) = (𝐹 “ ran 𝐺)
41	40	inteqi 4906	. . . 4 ⊢ ∩ ran (𝐹 ∘ 𝐺) = ∩ (𝐹 “ ran 𝐺)
42	39, 41, 40	3eltr3g 2852	. . 3 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦)) → ∩ (𝐹 “ ran 𝐺) ∈ (𝐹 “ ran 𝐺))
43		fnfvima 7179	. . 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 1373	. 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 482	. . . . . 6 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → 𝐴 ∈ Fin^III)
46	6, 7	sstrid 3945	. . . . . 6 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → (𝐹 “ ran 𝐺) ⊆ 𝒫 𝐴)
47		incom 4161	. . . . . . . . 9 ⊢ (dom 𝐹 ∩ ran 𝐺) = (ran 𝐺 ∩ dom 𝐹)
48		frn 6669	. . . . . . . . . . . 12 ⊢ (𝐺:ω⟶𝒫 𝐴 → ran 𝐺 ⊆ 𝒫 𝐴)
49	48	adantl 481	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → ran 𝐺 ⊆ 𝒫 𝐴)
50	3	fdmd 6672	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → dom 𝐹 = 𝒫 𝐴)
51	49, 50	sseqtrrd 3971	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → ran 𝐺 ⊆ dom 𝐹)
52		dfss2 3919	. . . . . . . . . 10 ⊢ (ran 𝐺 ⊆ dom 𝐹 ↔ (ran 𝐺 ∩ dom 𝐹) = ran 𝐺)
53	51, 52	sylib 218	. . . . . . . . 9 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → (ran 𝐺 ∩ dom 𝐹) = ran 𝐺)
54	47, 53	eqtrid 2783	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → (dom 𝐹 ∩ ran 𝐺) = ran 𝐺)
55		fdm 6671	. . . . . . . . . . 11 ⊢ (𝐺:ω⟶𝒫 𝐴 → dom 𝐺 = ω)
56	55	adantl 481	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → dom 𝐺 = ω)
57		peano1 7831	. . . . . . . . . . 11 ⊢ ∅ ∈ ω
58		ne0i 4293	. . . . . . . . . . 11 ⊢ (∅ ∈ ω → ω ≠ ∅)
59	57, 58	mp1i 13	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → ω ≠ ∅)
60	56, 59	eqnetrd 2999	. . . . . . . . 9 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → dom 𝐺 ≠ ∅)
61		dm0rn0 5873	. . . . . . . . . 10 ⊢ (dom 𝐺 = ∅ ↔ ran 𝐺 = ∅)
62	61	necon3bii 2984	. . . . . . . . 9 ⊢ (dom 𝐺 ≠ ∅ ↔ ran 𝐺 ≠ ∅)
63	60, 62	sylib 218	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → ran 𝐺 ≠ ∅)
64	54, 63	eqnetrd 2999	. . . . . . 7 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → (dom 𝐹 ∩ ran 𝐺) ≠ ∅)
65		imadisj 6039	. . . . . . . 8 ⊢ ((𝐹 “ ran 𝐺) = ∅ ↔ (dom 𝐹 ∩ ran 𝐺) = ∅)
66	65	necon3bii 2984	. . . . . . 7 ⊢ ((𝐹 “ ran 𝐺) ≠ ∅ ↔ (dom 𝐹 ∩ ran 𝐺) ≠ ∅)
67	64, 66	sylibr 234	. . . . . 6 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → (𝐹 “ ran 𝐺) ≠ ∅)
68	1	isf34lem5 10290	. . . . . 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 10287	. . . . . . 7 ⊢ ((𝐴 ∈ Fin^III ∧ ran 𝐺 ⊆ 𝒫 𝐴) → (𝐹 “ (𝐹 “ ran 𝐺)) = ran 𝐺)
71	45, 49, 70	syl2anc 584	. . . . . 6 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → (𝐹 “ (𝐹 “ ran 𝐺)) = ran 𝐺)
72	71	unieqd 4876	. . . . 5 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → ∪ (𝐹 “ (𝐹 “ ran 𝐺)) = ∪ ran 𝐺)
73	69, 72	eqtrd 2771	. . . 4 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → (𝐹‘∩ (𝐹 “ ran 𝐺)) = ∪ ran 𝐺)
74	73, 71	eleq12d 2830	. . 3 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → ((𝐹‘∩ (𝐹 “ ran 𝐺)) ∈ (𝐹 “ (𝐹 “ ran 𝐺)) ↔ ∪ ran 𝐺 ∈ ran 𝐺))
75	74	3adant3 1132	. 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 232	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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ≠ wne 2932 ∀wral 3051 ∖ cdif 3898 ∩ cin 3900 ⊆ wss 3901 ∅c0 4285 𝒫 cpw 4554 ∪ cuni 4863 ∩ cint 4902 ↦ cmpt 5179 dom cdm 5624 ran crn 5625 “ cima 5627 ∘ ccom 5628 suc csuc 6319 Fn wfn 6487 ⟶wf 6488 ‘cfv 6492 ωcom 7808 Fin^IIIcfin3 10193

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-rpss 7668 df-om 7809 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-er 8635 df-en 8886 df-dom 8887 df-sdom 8888 df-fin 8889 df-wdom 9472 df-card 9853 df-fin4 10199 df-fin3 10200

This theorem is referenced by: isf34lem6 10292 fin34i 10293

W3C validator