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Theorem isf34lem6 10372

Description: Lemma for isfin3-4 10374. (Contributed by Stefan O'Rear, 7-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)

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

**Assertion**
Ref	Expression
isf34lem6	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Fin^III ↔ ∀𝑓 ∈ (𝒫 𝐴 ↑_m ω)(∀𝑦 ∈ ω (𝑓‘𝑦) ⊆ (𝑓‘suc 𝑦) → ∪ ran 𝑓 ∈ ran 𝑓)))

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

Proof of Theorem isf34lem6 Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elmapi 8840	. . . 4 ⊢ (𝑓 ∈ (𝒫 𝐴 ↑_m ω) → 𝑓:ω⟶𝒫 𝐴)
2		compss.a	. . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥))
3	2	isf34lem7 10371	. . . . 5 ⊢ ((𝐴 ∈ Fin^III ∧ 𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝑓‘𝑦) ⊆ (𝑓‘suc 𝑦)) → ∪ ran 𝑓 ∈ ran 𝑓)
4	3	3expia 1122	. . . 4 ⊢ ((𝐴 ∈ Fin^III ∧ 𝑓:ω⟶𝒫 𝐴) → (∀𝑦 ∈ ω (𝑓‘𝑦) ⊆ (𝑓‘suc 𝑦) → ∪ ran 𝑓 ∈ ran 𝑓))
5	1, 4	sylan2 594	. . 3 ⊢ ((𝐴 ∈ Fin^III ∧ 𝑓 ∈ (𝒫 𝐴 ↑_m ω)) → (∀𝑦 ∈ ω (𝑓‘𝑦) ⊆ (𝑓‘suc 𝑦) → ∪ ran 𝑓 ∈ ran 𝑓))
6	5	ralrimiva 3147	. 2 ⊢ (𝐴 ∈ Fin^III → ∀𝑓 ∈ (𝒫 𝐴 ↑_m ω)(∀𝑦 ∈ ω (𝑓‘𝑦) ⊆ (𝑓‘suc 𝑦) → ∪ ran 𝑓 ∈ ran 𝑓))
7		elmapex 8839	. . . . . . . . . . 11 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑_m ω) → (𝒫 𝐴 ∈ V ∧ ω ∈ V))
8	7	simpld 496	. . . . . . . . . 10 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑_m ω) → 𝒫 𝐴 ∈ V)
9		pwexb 7750	. . . . . . . . . 10 ⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
10	8, 9	sylibr 233	. . . . . . . . 9 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑_m ω) → 𝐴 ∈ V)
11	2	isf34lem2 10365	. . . . . . . . 9 ⊢ (𝐴 ∈ V → 𝐹:𝒫 𝐴⟶𝒫 𝐴)
12	10, 11	syl 17	. . . . . . . 8 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑_m ω) → 𝐹:𝒫 𝐴⟶𝒫 𝐴)
13		elmapi 8840	. . . . . . . 8 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑_m ω) → 𝑔:ω⟶𝒫 𝐴)
14		fco 6739	. . . . . . . 8 ⊢ ((𝐹:𝒫 𝐴⟶𝒫 𝐴 ∧ 𝑔:ω⟶𝒫 𝐴) → (𝐹 ∘ 𝑔):ω⟶𝒫 𝐴)
15	12, 13, 14	syl2anc 585	. . . . . . 7 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑_m ω) → (𝐹 ∘ 𝑔):ω⟶𝒫 𝐴)
16		elmapg 8830	. . . . . . . 8 ⊢ ((𝒫 𝐴 ∈ V ∧ ω ∈ V) → ((𝐹 ∘ 𝑔) ∈ (𝒫 𝐴 ↑_m ω) ↔ (𝐹 ∘ 𝑔):ω⟶𝒫 𝐴))
17	7, 16	syl 17	. . . . . . 7 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑_m ω) → ((𝐹 ∘ 𝑔) ∈ (𝒫 𝐴 ↑_m ω) ↔ (𝐹 ∘ 𝑔):ω⟶𝒫 𝐴))
18	15, 17	mpbird 257	. . . . . 6 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑_m ω) → (𝐹 ∘ 𝑔) ∈ (𝒫 𝐴 ↑_m ω))
19		fveq1 6888	. . . . . . . . . 10 ⊢ (𝑓 = (𝐹 ∘ 𝑔) → (𝑓‘𝑦) = ((𝐹 ∘ 𝑔)‘𝑦))
20		fveq1 6888	. . . . . . . . . 10 ⊢ (𝑓 = (𝐹 ∘ 𝑔) → (𝑓‘suc 𝑦) = ((𝐹 ∘ 𝑔)‘suc 𝑦))
21	19, 20	sseq12d 4015	. . . . . . . . 9 ⊢ (𝑓 = (𝐹 ∘ 𝑔) → ((𝑓‘𝑦) ⊆ (𝑓‘suc 𝑦) ↔ ((𝐹 ∘ 𝑔)‘𝑦) ⊆ ((𝐹 ∘ 𝑔)‘suc 𝑦)))
22	21	ralbidv 3178	. . . . . . . 8 ⊢ (𝑓 = (𝐹 ∘ 𝑔) → (∀𝑦 ∈ ω (𝑓‘𝑦) ⊆ (𝑓‘suc 𝑦) ↔ ∀𝑦 ∈ ω ((𝐹 ∘ 𝑔)‘𝑦) ⊆ ((𝐹 ∘ 𝑔)‘suc 𝑦)))
23		rneq 5934	. . . . . . . . . . 11 ⊢ (𝑓 = (𝐹 ∘ 𝑔) → ran 𝑓 = ran (𝐹 ∘ 𝑔))
24		rnco2 6250	. . . . . . . . . . 11 ⊢ ran (𝐹 ∘ 𝑔) = (𝐹 “ ran 𝑔)
25	23, 24	eqtrdi 2789	. . . . . . . . . 10 ⊢ (𝑓 = (𝐹 ∘ 𝑔) → ran 𝑓 = (𝐹 “ ran 𝑔))
26	25	unieqd 4922	. . . . . . . . 9 ⊢ (𝑓 = (𝐹 ∘ 𝑔) → ∪ ran 𝑓 = ∪ (𝐹 “ ran 𝑔))
27	26, 25	eleq12d 2828	. . . . . . . 8 ⊢ (𝑓 = (𝐹 ∘ 𝑔) → (∪ ran 𝑓 ∈ ran 𝑓 ↔ ∪ (𝐹 “ ran 𝑔) ∈ (𝐹 “ ran 𝑔)))
28	22, 27	imbi12d 345	. . . . . . 7 ⊢ (𝑓 = (𝐹 ∘ 𝑔) → ((∀𝑦 ∈ ω (𝑓‘𝑦) ⊆ (𝑓‘suc 𝑦) → ∪ ran 𝑓 ∈ ran 𝑓) ↔ (∀𝑦 ∈ ω ((𝐹 ∘ 𝑔)‘𝑦) ⊆ ((𝐹 ∘ 𝑔)‘suc 𝑦) → ∪ (𝐹 “ ran 𝑔) ∈ (𝐹 “ ran 𝑔))))
29	28	rspccv 3610	. . . . . 6 ⊢ (∀𝑓 ∈ (𝒫 𝐴 ↑_m ω)(∀𝑦 ∈ ω (𝑓‘𝑦) ⊆ (𝑓‘suc 𝑦) → ∪ ran 𝑓 ∈ ran 𝑓) → ((𝐹 ∘ 𝑔) ∈ (𝒫 𝐴 ↑_m ω) → (∀𝑦 ∈ ω ((𝐹 ∘ 𝑔)‘𝑦) ⊆ ((𝐹 ∘ 𝑔)‘suc 𝑦) → ∪ (𝐹 “ ran 𝑔) ∈ (𝐹 “ ran 𝑔))))
30	18, 29	syl5 34	. . . . 5 ⊢ (∀𝑓 ∈ (𝒫 𝐴 ↑_m ω)(∀𝑦 ∈ ω (𝑓‘𝑦) ⊆ (𝑓‘suc 𝑦) → ∪ ran 𝑓 ∈ ran 𝑓) → (𝑔 ∈ (𝒫 𝐴 ↑_m ω) → (∀𝑦 ∈ ω ((𝐹 ∘ 𝑔)‘𝑦) ⊆ ((𝐹 ∘ 𝑔)‘suc 𝑦) → ∪ (𝐹 “ ran 𝑔) ∈ (𝐹 “ ran 𝑔))))
31		sscon 4138	. . . . . . . . 9 ⊢ ((𝑔‘suc 𝑦) ⊆ (𝑔‘𝑦) → (𝐴 ∖ (𝑔‘𝑦)) ⊆ (𝐴 ∖ (𝑔‘suc 𝑦)))
32	13	ffvelcdmda 7084	. . . . . . . . . . . 12 ⊢ ((𝑔 ∈ (𝒫 𝐴 ↑_m ω) ∧ 𝑦 ∈ ω) → (𝑔‘𝑦) ∈ 𝒫 𝐴)
33	32	elpwid 4611	. . . . . . . . . . 11 ⊢ ((𝑔 ∈ (𝒫 𝐴 ↑_m ω) ∧ 𝑦 ∈ ω) → (𝑔‘𝑦) ⊆ 𝐴)
34	2	isf34lem1 10364	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ V ∧ (𝑔‘𝑦) ⊆ 𝐴) → (𝐹‘(𝑔‘𝑦)) = (𝐴 ∖ (𝑔‘𝑦)))
35	10, 33, 34	syl2an2r 684	. . . . . . . . . 10 ⊢ ((𝑔 ∈ (𝒫 𝐴 ↑_m ω) ∧ 𝑦 ∈ ω) → (𝐹‘(𝑔‘𝑦)) = (𝐴 ∖ (𝑔‘𝑦)))
36		peano2 7878	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ω → suc 𝑦 ∈ ω)
37		ffvelcdm 7081	. . . . . . . . . . . . 13 ⊢ ((𝑔:ω⟶𝒫 𝐴 ∧ suc 𝑦 ∈ ω) → (𝑔‘suc 𝑦) ∈ 𝒫 𝐴)
38	13, 36, 37	syl2an 597	. . . . . . . . . . . 12 ⊢ ((𝑔 ∈ (𝒫 𝐴 ↑_m ω) ∧ 𝑦 ∈ ω) → (𝑔‘suc 𝑦) ∈ 𝒫 𝐴)
39	38	elpwid 4611	. . . . . . . . . . 11 ⊢ ((𝑔 ∈ (𝒫 𝐴 ↑_m ω) ∧ 𝑦 ∈ ω) → (𝑔‘suc 𝑦) ⊆ 𝐴)
40	2	isf34lem1 10364	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ V ∧ (𝑔‘suc 𝑦) ⊆ 𝐴) → (𝐹‘(𝑔‘suc 𝑦)) = (𝐴 ∖ (𝑔‘suc 𝑦)))
41	10, 39, 40	syl2an2r 684	. . . . . . . . . 10 ⊢ ((𝑔 ∈ (𝒫 𝐴 ↑_m ω) ∧ 𝑦 ∈ ω) → (𝐹‘(𝑔‘suc 𝑦)) = (𝐴 ∖ (𝑔‘suc 𝑦)))
42	35, 41	sseq12d 4015	. . . . . . . . 9 ⊢ ((𝑔 ∈ (𝒫 𝐴 ↑_m ω) ∧ 𝑦 ∈ ω) → ((𝐹‘(𝑔‘𝑦)) ⊆ (𝐹‘(𝑔‘suc 𝑦)) ↔ (𝐴 ∖ (𝑔‘𝑦)) ⊆ (𝐴 ∖ (𝑔‘suc 𝑦))))
43	31, 42	imbitrrid 245	. . . . . . . 8 ⊢ ((𝑔 ∈ (𝒫 𝐴 ↑_m ω) ∧ 𝑦 ∈ ω) → ((𝑔‘suc 𝑦) ⊆ (𝑔‘𝑦) → (𝐹‘(𝑔‘𝑦)) ⊆ (𝐹‘(𝑔‘suc 𝑦))))
44		fvco3 6988	. . . . . . . . . 10 ⊢ ((𝑔:ω⟶𝒫 𝐴 ∧ 𝑦 ∈ ω) → ((𝐹 ∘ 𝑔)‘𝑦) = (𝐹‘(𝑔‘𝑦)))
45	13, 44	sylan 581	. . . . . . . . 9 ⊢ ((𝑔 ∈ (𝒫 𝐴 ↑_m ω) ∧ 𝑦 ∈ ω) → ((𝐹 ∘ 𝑔)‘𝑦) = (𝐹‘(𝑔‘𝑦)))
46		fvco3 6988	. . . . . . . . . 10 ⊢ ((𝑔:ω⟶𝒫 𝐴 ∧ suc 𝑦 ∈ ω) → ((𝐹 ∘ 𝑔)‘suc 𝑦) = (𝐹‘(𝑔‘suc 𝑦)))
47	13, 36, 46	syl2an 597	. . . . . . . . 9 ⊢ ((𝑔 ∈ (𝒫 𝐴 ↑_m ω) ∧ 𝑦 ∈ ω) → ((𝐹 ∘ 𝑔)‘suc 𝑦) = (𝐹‘(𝑔‘suc 𝑦)))
48	45, 47	sseq12d 4015	. . . . . . . 8 ⊢ ((𝑔 ∈ (𝒫 𝐴 ↑_m ω) ∧ 𝑦 ∈ ω) → (((𝐹 ∘ 𝑔)‘𝑦) ⊆ ((𝐹 ∘ 𝑔)‘suc 𝑦) ↔ (𝐹‘(𝑔‘𝑦)) ⊆ (𝐹‘(𝑔‘suc 𝑦))))
49	43, 48	sylibrd 259	. . . . . . 7 ⊢ ((𝑔 ∈ (𝒫 𝐴 ↑_m ω) ∧ 𝑦 ∈ ω) → ((𝑔‘suc 𝑦) ⊆ (𝑔‘𝑦) → ((𝐹 ∘ 𝑔)‘𝑦) ⊆ ((𝐹 ∘ 𝑔)‘suc 𝑦)))
50	49	ralimdva 3168	. . . . . 6 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑_m ω) → (∀𝑦 ∈ ω (𝑔‘suc 𝑦) ⊆ (𝑔‘𝑦) → ∀𝑦 ∈ ω ((𝐹 ∘ 𝑔)‘𝑦) ⊆ ((𝐹 ∘ 𝑔)‘suc 𝑦)))
51	12	ffnd 6716	. . . . . . . 8 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑_m ω) → 𝐹 Fn 𝒫 𝐴)
52		imassrn 6069	. . . . . . . . 9 ⊢ (𝐹 “ ran 𝑔) ⊆ ran 𝐹
53	12	frnd 6723	. . . . . . . . 9 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑_m ω) → ran 𝐹 ⊆ 𝒫 𝐴)
54	52, 53	sstrid 3993	. . . . . . . 8 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑_m ω) → (𝐹 “ ran 𝑔) ⊆ 𝒫 𝐴)
55		fnfvima 7232	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝒫 𝐴 ∧ (𝐹 “ ran 𝑔) ⊆ 𝒫 𝐴 ∧ ∪ (𝐹 “ ran 𝑔) ∈ (𝐹 “ ran 𝑔)) → (𝐹‘∪ (𝐹 “ ran 𝑔)) ∈ (𝐹 “ (𝐹 “ ran 𝑔)))
56	55	3expia 1122	. . . . . . . 8 ⊢ ((𝐹 Fn 𝒫 𝐴 ∧ (𝐹 “ ran 𝑔) ⊆ 𝒫 𝐴) → (∪ (𝐹 “ ran 𝑔) ∈ (𝐹 “ ran 𝑔) → (𝐹‘∪ (𝐹 “ ran 𝑔)) ∈ (𝐹 “ (𝐹 “ ran 𝑔))))
57	51, 54, 56	syl2anc 585	. . . . . . 7 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑_m ω) → (∪ (𝐹 “ ran 𝑔) ∈ (𝐹 “ ran 𝑔) → (𝐹‘∪ (𝐹 “ ran 𝑔)) ∈ (𝐹 “ (𝐹 “ ran 𝑔))))
58		incom 4201	. . . . . . . . . . . . 13 ⊢ (dom 𝐹 ∩ ran 𝑔) = (ran 𝑔 ∩ dom 𝐹)
59	13	frnd 6723	. . . . . . . . . . . . . . 15 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑_m ω) → ran 𝑔 ⊆ 𝒫 𝐴)
60	12	fdmd 6726	. . . . . . . . . . . . . . 15 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑_m ω) → dom 𝐹 = 𝒫 𝐴)
61	59, 60	sseqtrrd 4023	. . . . . . . . . . . . . 14 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑_m ω) → ran 𝑔 ⊆ dom 𝐹)
62		df-ss 3965	. . . . . . . . . . . . . 14 ⊢ (ran 𝑔 ⊆ dom 𝐹 ↔ (ran 𝑔 ∩ dom 𝐹) = ran 𝑔)
63	61, 62	sylib 217	. . . . . . . . . . . . 13 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑_m ω) → (ran 𝑔 ∩ dom 𝐹) = ran 𝑔)
64	58, 63	eqtrid 2785	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑_m ω) → (dom 𝐹 ∩ ran 𝑔) = ran 𝑔)
65	13	fdmd 6726	. . . . . . . . . . . . . 14 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑_m ω) → dom 𝑔 = ω)
66		peano1 7876	. . . . . . . . . . . . . . 15 ⊢ ∅ ∈ ω
67		ne0i 4334	. . . . . . . . . . . . . . 15 ⊢ (∅ ∈ ω → ω ≠ ∅)
68	66, 67	mp1i 13	. . . . . . . . . . . . . 14 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑_m ω) → ω ≠ ∅)
69	65, 68	eqnetrd 3009	. . . . . . . . . . . . 13 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑_m ω) → dom 𝑔 ≠ ∅)
70		dm0rn0 5923	. . . . . . . . . . . . . 14 ⊢ (dom 𝑔 = ∅ ↔ ran 𝑔 = ∅)
71	70	necon3bii 2994	. . . . . . . . . . . . 13 ⊢ (dom 𝑔 ≠ ∅ ↔ ran 𝑔 ≠ ∅)
72	69, 71	sylib 217	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑_m ω) → ran 𝑔 ≠ ∅)
73	64, 72	eqnetrd 3009	. . . . . . . . . . 11 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑_m ω) → (dom 𝐹 ∩ ran 𝑔) ≠ ∅)
74		imadisj 6077	. . . . . . . . . . . 12 ⊢ ((𝐹 “ ran 𝑔) = ∅ ↔ (dom 𝐹 ∩ ran 𝑔) = ∅)
75	74	necon3bii 2994	. . . . . . . . . . 11 ⊢ ((𝐹 “ ran 𝑔) ≠ ∅ ↔ (dom 𝐹 ∩ ran 𝑔) ≠ ∅)
76	73, 75	sylibr 233	. . . . . . . . . 10 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑_m ω) → (𝐹 “ ran 𝑔) ≠ ∅)
77	2	isf34lem4 10369	. . . . . . . . . 10 ⊢ ((𝐴 ∈ V ∧ ((𝐹 “ ran 𝑔) ⊆ 𝒫 𝐴 ∧ (𝐹 “ ran 𝑔) ≠ ∅)) → (𝐹‘∪ (𝐹 “ ran 𝑔)) = ∩ (𝐹 “ (𝐹 “ ran 𝑔)))
78	10, 54, 76, 77	syl12anc 836	. . . . . . . . 9 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑_m ω) → (𝐹‘∪ (𝐹 “ ran 𝑔)) = ∩ (𝐹 “ (𝐹 “ ran 𝑔)))
79	2	isf34lem3 10367	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ V ∧ ran 𝑔 ⊆ 𝒫 𝐴) → (𝐹 “ (𝐹 “ ran 𝑔)) = ran 𝑔)
80	10, 59, 79	syl2anc 585	. . . . . . . . . 10 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑_m ω) → (𝐹 “ (𝐹 “ ran 𝑔)) = ran 𝑔)
81	80	inteqd 4955	. . . . . . . . 9 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑_m ω) → ∩ (𝐹 “ (𝐹 “ ran 𝑔)) = ∩ ran 𝑔)
82	78, 81	eqtrd 2773	. . . . . . . 8 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑_m ω) → (𝐹‘∪ (𝐹 “ ran 𝑔)) = ∩ ran 𝑔)
83	82, 80	eleq12d 2828	. . . . . . 7 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑_m ω) → ((𝐹‘∪ (𝐹 “ ran 𝑔)) ∈ (𝐹 “ (𝐹 “ ran 𝑔)) ↔ ∩ ran 𝑔 ∈ ran 𝑔))
84	57, 83	sylibd 238	. . . . . 6 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑_m ω) → (∪ (𝐹 “ ran 𝑔) ∈ (𝐹 “ ran 𝑔) → ∩ ran 𝑔 ∈ ran 𝑔))
85	50, 84	imim12d 81	. . . . 5 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑_m ω) → ((∀𝑦 ∈ ω ((𝐹 ∘ 𝑔)‘𝑦) ⊆ ((𝐹 ∘ 𝑔)‘suc 𝑦) → ∪ (𝐹 “ ran 𝑔) ∈ (𝐹 “ ran 𝑔)) → (∀𝑦 ∈ ω (𝑔‘suc 𝑦) ⊆ (𝑔‘𝑦) → ∩ ran 𝑔 ∈ ran 𝑔)))
86	30, 85	sylcom 30	. . . 4 ⊢ (∀𝑓 ∈ (𝒫 𝐴 ↑_m ω)(∀𝑦 ∈ ω (𝑓‘𝑦) ⊆ (𝑓‘suc 𝑦) → ∪ ran 𝑓 ∈ ran 𝑓) → (𝑔 ∈ (𝒫 𝐴 ↑_m ω) → (∀𝑦 ∈ ω (𝑔‘suc 𝑦) ⊆ (𝑔‘𝑦) → ∩ ran 𝑔 ∈ ran 𝑔)))
87	86	ralrimiv 3146	. . 3 ⊢ (∀𝑓 ∈ (𝒫 𝐴 ↑_m ω)(∀𝑦 ∈ ω (𝑓‘𝑦) ⊆ (𝑓‘suc 𝑦) → ∪ ran 𝑓 ∈ ran 𝑓) → ∀𝑔 ∈ (𝒫 𝐴 ↑_m ω)(∀𝑦 ∈ ω (𝑔‘suc 𝑦) ⊆ (𝑔‘𝑦) → ∩ ran 𝑔 ∈ ran 𝑔))
88		isfin3-3 10360	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Fin^III ↔ ∀𝑔 ∈ (𝒫 𝐴 ↑_m ω)(∀𝑦 ∈ ω (𝑔‘suc 𝑦) ⊆ (𝑔‘𝑦) → ∩ ran 𝑔 ∈ ran 𝑔)))
89	87, 88	imbitrrid 245	. 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑓 ∈ (𝒫 𝐴 ↑_m ω)(∀𝑦 ∈ ω (𝑓‘𝑦) ⊆ (𝑓‘suc 𝑦) → ∪ ran 𝑓 ∈ ran 𝑓) → 𝐴 ∈ Fin^III))
90	6, 89	impbid2 225	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Fin^III ↔ ∀𝑓 ∈ (𝒫 𝐴 ↑_m ω)(∀𝑦 ∈ ω (𝑓‘𝑦) ⊆ (𝑓‘suc 𝑦) → ∪ ran 𝑓 ∈ ran 𝑓)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 Vcvv 3475 ∖ cdif 3945 ∩ cin 3947 ⊆ wss 3948 ∅c0 4322 𝒫 cpw 4602 ∪ cuni 4908 ∩ cint 4950 ↦ cmpt 5231 dom cdm 5676 ran crn 5677 “ cima 5679 ∘ ccom 5680 suc csuc 6364 Fn wfn 6536 ⟶wf 6537 ‘cfv 6541 (class class class)co 7406 ωcom 7852 ↑_m cmap 8817 Fin^IIIcfin3 10273

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 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722

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 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-isom 6550 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-rpss 7710 df-om 7853 df-1st 7972 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-seqom 8445 df-1o 8463 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-wdom 9557 df-card 9931 df-fin4 10279 df-fin3 10280

This theorem is referenced by: isfin3-4 10374

W3C validator