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Theorem isf34lem7 10324

Description: Lemma for isfin3-4 10327. (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 10318	. . . . . 6 ⊢ (𝐴 ∈ Fin^III → 𝐹:𝒫 𝐴⟶𝒫 𝐴)
3	2	adantr 481	. . . . 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 6674	. . 3 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦)) → 𝐹 Fn 𝒫 𝐴)
6		imassrn 6029	. . . 4 ⊢ (𝐹 “ ran 𝐺) ⊆ ran 𝐹
7	3	frnd 6681	. . . . 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 3958	. . 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 6697	. . . . . . 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 4103	. . . . . . . 8 ⊢ ((𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦) → (𝐴 ∖ (𝐺‘suc 𝑦)) ⊆ (𝐴 ∖ (𝐺‘𝑦)))
15		simpr 485	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → 𝐺:ω⟶𝒫 𝐴)
16		peano2 7832	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ω → suc 𝑦 ∈ ω)
17		fvco3 6945	. . . . . . . . . . 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 765	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → 𝐴 ∈ Fin^III)
20		ffvelcdm 7037	. . . . . . . . . . . . 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 4574	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → (𝐺‘suc 𝑦) ⊆ 𝐴)
23	1	isf34lem1 10317	. . . . . . . . . . 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 6945	. . . . . . . . . . 11 ⊢ ((𝐺:ω⟶𝒫 𝐴 ∧ 𝑦 ∈ ω) → ((𝐹 ∘ 𝐺)‘𝑦) = (𝐹‘(𝐺‘𝑦)))
27	26	adantll 712	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → ((𝐹 ∘ 𝐺)‘𝑦) = (𝐹‘(𝐺‘𝑦)))
28		ffvelcdm 7037	. . . . . . . . . . . . 13 ⊢ ((𝐺:ω⟶𝒫 𝐴 ∧ 𝑦 ∈ ω) → (𝐺‘𝑦) ∈ 𝒫 𝐴)
29	28	adantll 712	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → (𝐺‘𝑦) ∈ 𝒫 𝐴)
30	29	elpwid 4574	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) ∧ 𝑦 ∈ ω) → (𝐺‘𝑦) ⊆ 𝐴)
31	1	isf34lem1 10317	. . . . . . . . . . 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 3980	. . . . . . . 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 3160	. . . . . 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 10314	. . . . 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 1371	. . . 4 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦)) → ∩ ran (𝐹 ∘ 𝐺) ∈ ran (𝐹 ∘ 𝐺))
40		rnco2 6210	. . . . 5 ⊢ ran (𝐹 ∘ 𝐺) = (𝐹 “ ran 𝐺)
41	40	inteqi 4916	. . . 4 ⊢ ∩ ran (𝐹 ∘ 𝐺) = ∩ (𝐹 “ ran 𝐺)
42	39, 41, 40	3eltr3g 2848	. . 3 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦)) → ∩ (𝐹 “ ran 𝐺) ∈ (𝐹 “ ran 𝐺))
43		fnfvima 7188	. . 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 1371	. 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 483	. . . . . 6 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → 𝐴 ∈ Fin^III)
46	6, 7	sstrid 3958	. . . . . 6 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → (𝐹 “ ran 𝐺) ⊆ 𝒫 𝐴)
47		incom 4166	. . . . . . . . 9 ⊢ (dom 𝐹 ∩ ran 𝐺) = (ran 𝐺 ∩ dom 𝐹)
48		frn 6680	. . . . . . . . . . . 12 ⊢ (𝐺:ω⟶𝒫 𝐴 → ran 𝐺 ⊆ 𝒫 𝐴)
49	48	adantl 482	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → ran 𝐺 ⊆ 𝒫 𝐴)
50	3	fdmd 6684	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → dom 𝐹 = 𝒫 𝐴)
51	49, 50	sseqtrrd 3988	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → ran 𝐺 ⊆ dom 𝐹)
52		df-ss 3930	. . . . . . . . . 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 2783	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → (dom 𝐹 ∩ ran 𝐺) = ran 𝐺)
55		fdm 6682	. . . . . . . . . . 11 ⊢ (𝐺:ω⟶𝒫 𝐴 → dom 𝐺 = ω)
56	55	adantl 482	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → dom 𝐺 = ω)
57		peano1 7830	. . . . . . . . . . 11 ⊢ ∅ ∈ ω
58		ne0i 4299	. . . . . . . . . . 11 ⊢ (∅ ∈ ω → ω ≠ ∅)
59	57, 58	mp1i 13	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → ω ≠ ∅)
60	56, 59	eqnetrd 3007	. . . . . . . . 9 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → dom 𝐺 ≠ ∅)
61		dm0rn0 5885	. . . . . . . . . 10 ⊢ (dom 𝐺 = ∅ ↔ ran 𝐺 = ∅)
62	61	necon3bii 2992	. . . . . . . . 9 ⊢ (dom 𝐺 ≠ ∅ ↔ ran 𝐺 ≠ ∅)
63	60, 62	sylib 217	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → ran 𝐺 ≠ ∅)
64	54, 63	eqnetrd 3007	. . . . . . 7 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → (dom 𝐹 ∩ ran 𝐺) ≠ ∅)
65		imadisj 6037	. . . . . . . 8 ⊢ ((𝐹 “ ran 𝐺) = ∅ ↔ (dom 𝐹 ∩ ran 𝐺) = ∅)
66	65	necon3bii 2992	. . . . . . 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 10323	. . . . . 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 835	. . . . 5 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐺:ω⟶𝒫 𝐴) → (𝐹‘∩ (𝐹 “ ran 𝐺)) = ∪ (𝐹 “ (𝐹 “ ran 𝐺)))
70	1	isf34lem3 10320	. . . . . . 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 4884	. . . . 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 2826	. . 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 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 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2939 ∀wral 3060 ∖ cdif 3910 ∩ cin 3912 ⊆ wss 3913 ∅c0 4287 𝒫 cpw 4565 ∪ cuni 4870 ∩ cint 4912 ↦ cmpt 5193 dom cdm 5638 ran crn 5639 “ cima 5641 ∘ ccom 5642 suc csuc 6324 Fn wfn 6496 ⟶wf 6497 ‘cfv 6501 ωcom 7807 Fin^IIIcfin3 10226

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3448 df-sbc 3743 df-csb 3859 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-rpss 7665 df-om 7808 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-wdom 9510 df-card 9884 df-fin4 10232 df-fin3 10233

This theorem is referenced by: isf34lem6 10325 fin34i 10326

W3C validator