Theorem isf34lem6 10333

Description: Lemma for isfin3-4 10335. (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 ∈ FinIII ↔ ∀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 8822	. . . 4 ⊢ (𝑓 ∈ (𝒫 𝐴 ↑m ω) → 𝑓:ω⟶𝒫 𝐴)
2		compss.a	. . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥))
3	2	isf34lem7 10332	. . . . 5 ⊢ ((𝐴 ∈ FinIII ∧ 𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝑓‘𝑦) ⊆ (𝑓‘suc 𝑦)) → ∪ ran 𝑓 ∈ ran 𝑓)
4	3	3expia 1121	. . . 4 ⊢ ((𝐴 ∈ FinIII ∧ 𝑓:ω⟶𝒫 𝐴) → (∀𝑦 ∈ ω (𝑓‘𝑦) ⊆ (𝑓‘suc 𝑦) → ∪ ran 𝑓 ∈ ran 𝑓))
5	1, 4	sylan2 593	. . 3 ⊢ ((𝐴 ∈ FinIII ∧ 𝑓 ∈ (𝒫 𝐴 ↑m ω)) → (∀𝑦 ∈ ω (𝑓‘𝑦) ⊆ (𝑓‘suc 𝑦) → ∪ ran 𝑓 ∈ ran 𝑓))
6	5	ralrimiva 3125	. 2 ⊢ (𝐴 ∈ FinIII → ∀𝑓 ∈ (𝒫 𝐴 ↑m ω)(∀𝑦 ∈ ω (𝑓‘𝑦) ⊆ (𝑓‘suc 𝑦) → ∪ ran 𝑓 ∈ ran 𝑓))
7		elmapex 8821	. . . . . . . . . . 11 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → (𝒫 𝐴 ∈ V ∧ ω ∈ V))
8	7	simpld 494	. . . . . . . . . 10 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → 𝒫 𝐴 ∈ V)
9		pwexb 7742	. . . . . . . . . 10 ⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
10	8, 9	sylibr 234	. . . . . . . . 9 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → 𝐴 ∈ V)
11	2	isf34lem2 10326	. . . . . . . . 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 8822	. . . . . . . 8 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → 𝑔:ω⟶𝒫 𝐴)
14		fco 6712	. . . . . . . 8 ⊢ ((𝐹:𝒫 𝐴⟶𝒫 𝐴 ∧ 𝑔:ω⟶𝒫 𝐴) → (𝐹 ∘ 𝑔):ω⟶𝒫 𝐴)
15	12, 13, 14	syl2anc 584	. . . . . . 7 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → (𝐹 ∘ 𝑔):ω⟶𝒫 𝐴)
16		elmapg 8812	. . . . . . . 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 6857	. . . . . . . . . 10 ⊢ (𝑓 = (𝐹 ∘ 𝑔) → (𝑓‘𝑦) = ((𝐹 ∘ 𝑔)‘𝑦))
20		fveq1 6857	. . . . . . . . . 10 ⊢ (𝑓 = (𝐹 ∘ 𝑔) → (𝑓‘suc 𝑦) = ((𝐹 ∘ 𝑔)‘suc 𝑦))
21	19, 20	sseq12d 3980	. . . . . . . . 9 ⊢ (𝑓 = (𝐹 ∘ 𝑔) → ((𝑓‘𝑦) ⊆ (𝑓‘suc 𝑦) ↔ ((𝐹 ∘ 𝑔)‘𝑦) ⊆ ((𝐹 ∘ 𝑔)‘suc 𝑦)))
22	21	ralbidv 3156	. . . . . . . 8 ⊢ (𝑓 = (𝐹 ∘ 𝑔) → (∀𝑦 ∈ ω (𝑓‘𝑦) ⊆ (𝑓‘suc 𝑦) ↔ ∀𝑦 ∈ ω ((𝐹 ∘ 𝑔)‘𝑦) ⊆ ((𝐹 ∘ 𝑔)‘suc 𝑦)))
23		rneq 5900	. . . . . . . . . . 11 ⊢ (𝑓 = (𝐹 ∘ 𝑔) → ran 𝑓 = ran (𝐹 ∘ 𝑔))
24		rnco2 6226	. . . . . . . . . . 11 ⊢ ran (𝐹 ∘ 𝑔) = (𝐹 “ ran 𝑔)
25	23, 24	eqtrdi 2780	. . . . . . . . . 10 ⊢ (𝑓 = (𝐹 ∘ 𝑔) → ran 𝑓 = (𝐹 “ ran 𝑔))
26	25	unieqd 4884	. . . . . . . . 9 ⊢ (𝑓 = (𝐹 ∘ 𝑔) → ∪ ran 𝑓 = ∪ (𝐹 “ ran 𝑔))
27	26, 25	eleq12d 2822	. . . . . . . 8 ⊢ (𝑓 = (𝐹 ∘ 𝑔) → (∪ ran 𝑓 ∈ ran 𝑓 ↔ ∪ (𝐹 “ ran 𝑔) ∈ (𝐹 “ ran 𝑔)))
28	22, 27	imbi12d 344	. . . . . . 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 3585	. . . . . 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 4106	. . . . . . . . 9 ⊢ ((𝑔‘suc 𝑦) ⊆ (𝑔‘𝑦) → (𝐴 ∖ (𝑔‘𝑦)) ⊆ (𝐴 ∖ (𝑔‘suc 𝑦)))
32	13	ffvelcdmda 7056	. . . . . . . . . . . 12 ⊢ ((𝑔 ∈ (𝒫 𝐴 ↑m ω) ∧ 𝑦 ∈ ω) → (𝑔‘𝑦) ∈ 𝒫 𝐴)
33	32	elpwid 4572	. . . . . . . . . . 11 ⊢ ((𝑔 ∈ (𝒫 𝐴 ↑m ω) ∧ 𝑦 ∈ ω) → (𝑔‘𝑦) ⊆ 𝐴)
34	2	isf34lem1 10325	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ V ∧ (𝑔‘𝑦) ⊆ 𝐴) → (𝐹‘(𝑔‘𝑦)) = (𝐴 ∖ (𝑔‘𝑦)))
35	10, 33, 34	syl2an2r 685	. . . . . . . . . 10 ⊢ ((𝑔 ∈ (𝒫 𝐴 ↑m ω) ∧ 𝑦 ∈ ω) → (𝐹‘(𝑔‘𝑦)) = (𝐴 ∖ (𝑔‘𝑦)))
36		peano2 7866	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ω → suc 𝑦 ∈ ω)
37		ffvelcdm 7053	. . . . . . . . . . . . 13 ⊢ ((𝑔:ω⟶𝒫 𝐴 ∧ suc 𝑦 ∈ ω) → (𝑔‘suc 𝑦) ∈ 𝒫 𝐴)
38	13, 36, 37	syl2an 596	. . . . . . . . . . . 12 ⊢ ((𝑔 ∈ (𝒫 𝐴 ↑m ω) ∧ 𝑦 ∈ ω) → (𝑔‘suc 𝑦) ∈ 𝒫 𝐴)
39	38	elpwid 4572	. . . . . . . . . . 11 ⊢ ((𝑔 ∈ (𝒫 𝐴 ↑m ω) ∧ 𝑦 ∈ ω) → (𝑔‘suc 𝑦) ⊆ 𝐴)
40	2	isf34lem1 10325	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ V ∧ (𝑔‘suc 𝑦) ⊆ 𝐴) → (𝐹‘(𝑔‘suc 𝑦)) = (𝐴 ∖ (𝑔‘suc 𝑦)))
41	10, 39, 40	syl2an2r 685	. . . . . . . . . 10 ⊢ ((𝑔 ∈ (𝒫 𝐴 ↑m ω) ∧ 𝑦 ∈ ω) → (𝐹‘(𝑔‘suc 𝑦)) = (𝐴 ∖ (𝑔‘suc 𝑦)))
42	35, 41	sseq12d 3980	. . . . . . . . 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 246	. . . . . . . 8 ⊢ ((𝑔 ∈ (𝒫 𝐴 ↑m ω) ∧ 𝑦 ∈ ω) → ((𝑔‘suc 𝑦) ⊆ (𝑔‘𝑦) → (𝐹‘(𝑔‘𝑦)) ⊆ (𝐹‘(𝑔‘suc 𝑦))))
44		fvco3 6960	. . . . . . . . . 10 ⊢ ((𝑔:ω⟶𝒫 𝐴 ∧ 𝑦 ∈ ω) → ((𝐹 ∘ 𝑔)‘𝑦) = (𝐹‘(𝑔‘𝑦)))
45	13, 44	sylan 580	. . . . . . . . 9 ⊢ ((𝑔 ∈ (𝒫 𝐴 ↑m ω) ∧ 𝑦 ∈ ω) → ((𝐹 ∘ 𝑔)‘𝑦) = (𝐹‘(𝑔‘𝑦)))
46		fvco3 6960	. . . . . . . . . 10 ⊢ ((𝑔:ω⟶𝒫 𝐴 ∧ suc 𝑦 ∈ ω) → ((𝐹 ∘ 𝑔)‘suc 𝑦) = (𝐹‘(𝑔‘suc 𝑦)))
47	13, 36, 46	syl2an 596	. . . . . . . . 9 ⊢ ((𝑔 ∈ (𝒫 𝐴 ↑m ω) ∧ 𝑦 ∈ ω) → ((𝐹 ∘ 𝑔)‘suc 𝑦) = (𝐹‘(𝑔‘suc 𝑦)))
48	45, 47	sseq12d 3980	. . . . . . . 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 3145	. . . . . 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 6689	. . . . . . . 8 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → 𝐹 Fn 𝒫 𝐴)
52		imassrn 6042	. . . . . . . . 9 ⊢ (𝐹 “ ran 𝑔) ⊆ ran 𝐹
53	12	frnd 6696	. . . . . . . . 9 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → ran 𝐹 ⊆ 𝒫 𝐴)
54	52, 53	sstrid 3958	. . . . . . . 8 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → (𝐹 “ ran 𝑔) ⊆ 𝒫 𝐴)
55		fnfvima 7207	. . . . . . . . 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 1121	. . . . . . . 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 584	. . . . . . 7 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → (∪ (𝐹 “ ran 𝑔) ∈ (𝐹 “ ran 𝑔) → (𝐹‘∪ (𝐹 “ ran 𝑔)) ∈ (𝐹 “ (𝐹 “ ran 𝑔))))
58		incom 4172	. . . . . . . . . . . . 13 ⊢ (dom 𝐹 ∩ ran 𝑔) = (ran 𝑔 ∩ dom 𝐹)
59	13	frnd 6696	. . . . . . . . . . . . . . 15 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → ran 𝑔 ⊆ 𝒫 𝐴)
60	12	fdmd 6698	. . . . . . . . . . . . . . 15 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → dom 𝐹 = 𝒫 𝐴)
61	59, 60	sseqtrrd 3984	. . . . . . . . . . . . . 14 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → ran 𝑔 ⊆ dom 𝐹)
62		dfss2 3932	. . . . . . . . . . . . . 14 ⊢ (ran 𝑔 ⊆ dom 𝐹 ↔ (ran 𝑔 ∩ dom 𝐹) = ran 𝑔)
63	61, 62	sylib 218	. . . . . . . . . . . . 13 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → (ran 𝑔 ∩ dom 𝐹) = ran 𝑔)
64	58, 63	eqtrid 2776	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → (dom 𝐹 ∩ ran 𝑔) = ran 𝑔)
65	13	fdmd 6698	. . . . . . . . . . . . . 14 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → dom 𝑔 = ω)
66		peano1 7865	. . . . . . . . . . . . . . 15 ⊢ ∅ ∈ ω
67		ne0i 4304	. . . . . . . . . . . . . . 15 ⊢ (∅ ∈ ω → ω ≠ ∅)
68	66, 67	mp1i 13	. . . . . . . . . . . . . 14 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → ω ≠ ∅)
69	65, 68	eqnetrd 2992	. . . . . . . . . . . . 13 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → dom 𝑔 ≠ ∅)
70		dm0rn0 5888	. . . . . . . . . . . . . 14 ⊢ (dom 𝑔 = ∅ ↔ ran 𝑔 = ∅)
71	70	necon3bii 2977	. . . . . . . . . . . . 13 ⊢ (dom 𝑔 ≠ ∅ ↔ ran 𝑔 ≠ ∅)
72	69, 71	sylib 218	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → ran 𝑔 ≠ ∅)
73	64, 72	eqnetrd 2992	. . . . . . . . . . 11 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → (dom 𝐹 ∩ ran 𝑔) ≠ ∅)
74		imadisj 6051	. . . . . . . . . . . 12 ⊢ ((𝐹 “ ran 𝑔) = ∅ ↔ (dom 𝐹 ∩ ran 𝑔) = ∅)
75	74	necon3bii 2977	. . . . . . . . . . 11 ⊢ ((𝐹 “ ran 𝑔) ≠ ∅ ↔ (dom 𝐹 ∩ ran 𝑔) ≠ ∅)
76	73, 75	sylibr 234	. . . . . . . . . 10 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → (𝐹 “ ran 𝑔) ≠ ∅)
77	2	isf34lem4 10330	. . . . . . . . . 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 10328	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ V ∧ ran 𝑔 ⊆ 𝒫 𝐴) → (𝐹 “ (𝐹 “ ran 𝑔)) = ran 𝑔)
80	10, 59, 79	syl2anc 584	. . . . . . . . . 10 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → (𝐹 “ (𝐹 “ ran 𝑔)) = ran 𝑔)
81	80	inteqd 4915	. . . . . . . . 9 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → ∩ (𝐹 “ (𝐹 “ ran 𝑔)) = ∩ ran 𝑔)
82	78, 81	eqtrd 2764	. . . . . . . 8 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → (𝐹‘∪ (𝐹 “ ran 𝑔)) = ∩ ran 𝑔)
83	82, 80	eleq12d 2822	. . . . . . 7 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → ((𝐹‘∪ (𝐹 “ ran 𝑔)) ∈ (𝐹 “ (𝐹 “ ran 𝑔)) ↔ ∩ ran 𝑔 ∈ ran 𝑔))
84	57, 83	sylibd 239	. . . . . 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 3124	. . 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 10321	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinIII ↔ ∀𝑔 ∈ (𝒫 𝐴 ↑m ω)(∀𝑦 ∈ ω (𝑔‘suc 𝑦) ⊆ (𝑔‘𝑦) → ∩ ran 𝑔 ∈ ran 𝑔)))
89	87, 88	imbitrrid 246	. 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑓 ∈ (𝒫 𝐴 ↑m ω)(∀𝑦 ∈ ω (𝑓‘𝑦) ⊆ (𝑓‘suc 𝑦) → ∪ ran 𝑓 ∈ ran 𝑓) → 𝐴 ∈ FinIII))
90	6, 89	impbid2 226	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinIII ↔ ∀𝑓 ∈ (𝒫 𝐴 ↑m ω)(∀𝑦 ∈ ω (𝑓‘𝑦) ⊆ (𝑓‘suc 𝑦) → ∪ ran 𝑓 ∈ ran 𝑓)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  Vcvv 3447   ∖ cdif 3911   ∩ cin 3913   ⊆ wss 3914  ∅c0 4296  𝒫 cpw 4563  ∪ cuni 4871  ∩ cint 4910   ↦ cmpt 5188  dom cdm 5638  ran crn 5639   “ cima 5641   ∘ ccom 5642  suc csuc 6334   Fn wfn 6506  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387  ωcom 7842   ↑_m cmap 8799  Fin^IIIcfin3 10234

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-se 5592  df-we 5593  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 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-isom 6520  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-rpss 7699  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-seqom 8416  df-1o 8434  df-er 8671  df-map 8801  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-wdom 9518  df-card 9892  df-fin4 10240  df-fin3 10241

This theorem is referenced by:  isfin3-4  10335