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Theorem isf34lem7 10066

Description: Lemma for isfin3-4 10069. (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 10060	. . . . . 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 1130	. . . 4 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦)) → 𝐹:𝒫 𝐴⟶𝒫 𝐴)
5	4	ffnd 6585	. . 3 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦)) → 𝐹 Fn 𝒫 𝐴)
6		imassrn 5969	. . . 4 ⊢ (𝐹 “ ran 𝐺) ⊆ ran 𝐹
7	3	frnd 6592	. . . . 5 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → ran 𝐹 ⊆ 𝒫 𝐴)
8	7	3adant3 1130	. . . 4 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦)) → ran 𝐹 ⊆ 𝒫 𝐴)
9	6, 8	sstrid 3928	. . 3 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦)) → (𝐹 “ ran 𝐺) ⊆ 𝒫 𝐴)
10		simp1 1134	. . . . 5 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦)) → 𝐴 ∈ Fin^III)
11		fco 6608	. . . . . . 7 ⊢ ((𝐹:𝒫 𝐴⟶𝒫 𝐴 ∧ 𝐺:ω⟶𝒫 𝐴) → (𝐹 ∘ 𝐺):ω⟶𝒫 𝐴)
12	2, 11	sylan 579	. . . . . 6 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → (𝐹 ∘ 𝐺):ω⟶𝒫 𝐴)
13	12	3adant3 1130	. . . . 5 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦)) → (𝐹 ∘ 𝐺):ω⟶𝒫 𝐴)
14		sscon 4069	. . . . . . . 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 7711	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ω → suc 𝑦 ∈ ω)
17		fvco3 6849	. . . . . . . . . . 11 ⊢ ((𝐺:ω⟶𝒫 𝐴 ∧ suc 𝑦 ∈ ω) → ((𝐹 ∘ 𝐺)‘suc 𝑦) = (𝐹‘(𝐺‘suc 𝑦)))
18	15, 16, 17	syl2an 595	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → ((𝐹 ∘ 𝐺)‘suc 𝑦) = (𝐹‘(𝐺‘suc 𝑦)))
19		simpll 763	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → 𝐴 ∈ Fin^III)
20		ffvelrn 6941	. . . . . . . . . . . . 13 ⊢ ((𝐺:ω⟶𝒫 𝐴 ∧ suc 𝑦 ∈ ω) → (𝐺‘suc 𝑦) ∈ 𝒫 𝐴)
21	15, 16, 20	syl2an 595	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → (𝐺‘suc 𝑦) ∈ 𝒫 𝐴)
22	21	elpwid 4541	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → (𝐺‘suc 𝑦) ⊆ 𝐴)
23	1	isf34lem1 10059	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin^III ∧ (𝐺‘suc 𝑦) ⊆ 𝐴) → (𝐹‘(𝐺‘suc 𝑦)) = (𝐴 ∖ (𝐺‘suc 𝑦)))
24	19, 22, 23	syl2anc 583	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → (𝐹‘(𝐺‘suc 𝑦)) = (𝐴 ∖ (𝐺‘suc 𝑦)))
25	18, 24	eqtrd 2778	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → ((𝐹 ∘ 𝐺)‘suc 𝑦) = (𝐴 ∖ (𝐺‘suc 𝑦)))
26		fvco3 6849	. . . . . . . . . . 11 ⊢ ((𝐺:ω⟶𝒫 𝐴 ∧ 𝑦 ∈ ω) → ((𝐹 ∘ 𝐺)‘𝑦) = (𝐹‘(𝐺‘𝑦)))
27	26	adantll 710	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → ((𝐹 ∘ 𝐺)‘𝑦) = (𝐹‘(𝐺‘𝑦)))
28		ffvelrn 6941	. . . . . . . . . . . . 13 ⊢ ((𝐺:ω⟶𝒫 𝐴 ∧ 𝑦 ∈ ω) → (𝐺‘𝑦) ∈ 𝒫 𝐴)
29	28	adantll 710	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → (𝐺‘𝑦) ∈ 𝒫 𝐴)
30	29	elpwid 4541	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → (𝐺‘𝑦) ⊆ 𝐴)
31	1	isf34lem1 10059	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin^III ∧ (𝐺‘𝑦) ⊆ 𝐴) → (𝐹‘(𝐺‘𝑦)) = (𝐴 ∖ (𝐺‘𝑦)))
32	19, 30, 31	syl2anc 583	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → (𝐹‘(𝐺‘𝑦)) = (𝐴 ∖ (𝐺‘𝑦)))
33	27, 32	eqtrd 2778	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → ((𝐹 ∘ 𝐺)‘𝑦) = (𝐴 ∖ (𝐺‘𝑦)))
34	25, 33	sseq12d 3950	. . . . . . . 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	syl5ibr 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 3102	. . . . . 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 1115	. . . . 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 10056	. . . . 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 1369	. . . 4 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦)) → ∩ ran (𝐹 ∘ 𝐺) ∈ ran (𝐹 ∘ 𝐺))
40		rnco2 6146	. . . . 5 ⊢ ran (𝐹 ∘ 𝐺) = (𝐹 “ ran 𝐺)
41	40	inteqi 4880	. . . 4 ⊢ ∩ ran (𝐹 ∘ 𝐺) = ∩ (𝐹 “ ran 𝐺)
42	39, 41, 40	3eltr3g 2855	. . 3 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦)) → ∩ (𝐹 “ ran 𝐺) ∈ (𝐹 “ ran 𝐺))
43		fnfvima 7091	. . 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 1369	. 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 3928	. . . . . 6 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → (𝐹 “ ran 𝐺) ⊆ 𝒫 𝐴)
47		incom 4131	. . . . . . . . 9 ⊢ (dom 𝐹 ∩ ran 𝐺) = (ran 𝐺 ∩ dom 𝐹)
48		frn 6591	. . . . . . . . . . . 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 6595	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → dom 𝐹 = 𝒫 𝐴)
51	49, 50	sseqtrrd 3958	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → ran 𝐺 ⊆ dom 𝐹)
52		df-ss 3900	. . . . . . . . . 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 2790	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → (dom 𝐹 ∩ ran 𝐺) = ran 𝐺)
55		fdm 6593	. . . . . . . . . . 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 7710	. . . . . . . . . . 11 ⊢ ∅ ∈ ω
58		ne0i 4265	. . . . . . . . . . 11 ⊢ (∅ ∈ ω → ω ≠ ∅)
59	57, 58	mp1i 13	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → ω ≠ ∅)
60	56, 59	eqnetrd 3010	. . . . . . . . 9 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → dom 𝐺 ≠ ∅)
61		dm0rn0 5823	. . . . . . . . . 10 ⊢ (dom 𝐺 = ∅ ↔ ran 𝐺 = ∅)
62	61	necon3bii 2995	. . . . . . . . 9 ⊢ (dom 𝐺 ≠ ∅ ↔ ran 𝐺 ≠ ∅)
63	60, 62	sylib 217	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → ran 𝐺 ≠ ∅)
64	54, 63	eqnetrd 3010	. . . . . . 7 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → (dom 𝐹 ∩ ran 𝐺) ≠ ∅)
65		imadisj 5977	. . . . . . . 8 ⊢ ((𝐹 “ ran 𝐺) = ∅ ↔ (dom 𝐹 ∩ ran 𝐺) = ∅)
66	65	necon3bii 2995	. . . . . . 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 10065	. . . . . 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 833	. . . . 5 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → (𝐹‘∩ (𝐹 “ ran 𝐺)) = ∪ (𝐹 “ (𝐹 “ ran 𝐺)))
70	1	isf34lem3 10062	. . . . . . 7 ⊢ ((𝐴 ∈ Fin^III ∧ ran 𝐺 ⊆ 𝒫 𝐴) → (𝐹 “ (𝐹 “ ran 𝐺)) = ran 𝐺)
71	45, 49, 70	syl2anc 583	. . . . . 6 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → (𝐹 “ (𝐹 “ ran 𝐺)) = ran 𝐺)
72	71	unieqd 4850	. . . . 5 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → ∪ (𝐹 “ (𝐹 “ ran 𝐺)) = ∪ ran 𝐺)
73	69, 72	eqtrd 2778	. . . 4 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → (𝐹‘∩ (𝐹 “ ran 𝐺)) = ∪ ran 𝐺)
74	73, 71	eleq12d 2833	. . 3 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → ((𝐹‘∩ (𝐹 “ ran 𝐺)) ∈ (𝐹 “ (𝐹 “ ran 𝐺)) ↔ ∪ ran 𝐺 ∈ ran 𝐺))
75	74	3adant3 1130	. 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 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∀wral 3063 ∖ cdif 3880 ∩ cin 3882 ⊆ wss 3883 ∅c0 4253 𝒫 cpw 4530 ∪ cuni 4836 ∩ cint 4876 ↦ cmpt 5153 dom cdm 5580 ran crn 5581 “ cima 5583 ∘ ccom 5584 suc csuc 6253 Fn wfn 6413 ⟶wf 6414 ‘cfv 6418 ωcom 7687 Fin^IIIcfin3 9968

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-rpss 7554 df-om 7688 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-wdom 9254 df-card 9628 df-fin4 9974 df-fin3 9975

This theorem is referenced by: isf34lem6 10067 fin34i 10068

W3C validator