Theorem isf34lem6 10136

Description: Lemma for isfin3-4 10138. (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 8637	. . . 4 ⊢ (𝑓 ∈ (𝒫 𝐴 ↑m ω) → 𝑓:ω⟶𝒫 𝐴)
2		compss.a	. . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥))
3	2	isf34lem7 10135	. . . . 5 ⊢ ((𝐴 ∈ FinIII ∧ 𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝑓‘𝑦) ⊆ (𝑓‘suc 𝑦)) → ∪ ran 𝑓 ∈ ran 𝑓)
4	3	3expia 1120	. . . 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 3103	. 2 ⊢ (𝐴 ∈ FinIII → ∀𝑓 ∈ (𝒫 𝐴 ↑m ω)(∀𝑦 ∈ ω (𝑓‘𝑦) ⊆ (𝑓‘suc 𝑦) → ∪ ran 𝑓 ∈ ran 𝑓))
7		elmapex 8636	. . . . . . . . . . 11 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → (𝒫 𝐴 ∈ V ∧ ω ∈ V))
8	7	simpld 495	. . . . . . . . . 10 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → 𝒫 𝐴 ∈ V)
9		pwexb 7616	. . . . . . . . . 10 ⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
10	8, 9	sylibr 233	. . . . . . . . 9 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → 𝐴 ∈ V)
11	2	isf34lem2 10129	. . . . . . . . 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 8637	. . . . . . . 8 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → 𝑔:ω⟶𝒫 𝐴)
14		fco 6624	. . . . . . . 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 8628	. . . . . . . 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 256	. . . . . 6 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → (𝐹 ∘ 𝑔) ∈ (𝒫 𝐴 ↑m ω))
19		fveq1 6773	. . . . . . . . . 10 ⊢ (𝑓 = (𝐹 ∘ 𝑔) → (𝑓‘𝑦) = ((𝐹 ∘ 𝑔)‘𝑦))
20		fveq1 6773	. . . . . . . . . 10 ⊢ (𝑓 = (𝐹 ∘ 𝑔) → (𝑓‘suc 𝑦) = ((𝐹 ∘ 𝑔)‘suc 𝑦))
21	19, 20	sseq12d 3954	. . . . . . . . 9 ⊢ (𝑓 = (𝐹 ∘ 𝑔) → ((𝑓‘𝑦) ⊆ (𝑓‘suc 𝑦) ↔ ((𝐹 ∘ 𝑔)‘𝑦) ⊆ ((𝐹 ∘ 𝑔)‘suc 𝑦)))
22	21	ralbidv 3112	. . . . . . . 8 ⊢ (𝑓 = (𝐹 ∘ 𝑔) → (∀𝑦 ∈ ω (𝑓‘𝑦) ⊆ (𝑓‘suc 𝑦) ↔ ∀𝑦 ∈ ω ((𝐹 ∘ 𝑔)‘𝑦) ⊆ ((𝐹 ∘ 𝑔)‘suc 𝑦)))
23		rneq 5845	. . . . . . . . . . 11 ⊢ (𝑓 = (𝐹 ∘ 𝑔) → ran 𝑓 = ran (𝐹 ∘ 𝑔))
24		rnco2 6157	. . . . . . . . . . 11 ⊢ ran (𝐹 ∘ 𝑔) = (𝐹 “ ran 𝑔)
25	23, 24	eqtrdi 2794	. . . . . . . . . 10 ⊢ (𝑓 = (𝐹 ∘ 𝑔) → ran 𝑓 = (𝐹 “ ran 𝑔))
26	25	unieqd 4853	. . . . . . . . 9 ⊢ (𝑓 = (𝐹 ∘ 𝑔) → ∪ ran 𝑓 = ∪ (𝐹 “ ran 𝑔))
27	26, 25	eleq12d 2833	. . . . . . . 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 3558	. . . . . 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 4073	. . . . . . . . 9 ⊢ ((𝑔‘suc 𝑦) ⊆ (𝑔‘𝑦) → (𝐴 ∖ (𝑔‘𝑦)) ⊆ (𝐴 ∖ (𝑔‘suc 𝑦)))
32	13	ffvelrnda 6961	. . . . . . . . . . . 12 ⊢ ((𝑔 ∈ (𝒫 𝐴 ↑m ω) ∧ 𝑦 ∈ ω) → (𝑔‘𝑦) ∈ 𝒫 𝐴)
33	32	elpwid 4544	. . . . . . . . . . 11 ⊢ ((𝑔 ∈ (𝒫 𝐴 ↑m ω) ∧ 𝑦 ∈ ω) → (𝑔‘𝑦) ⊆ 𝐴)
34	2	isf34lem1 10128	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ V ∧ (𝑔‘𝑦) ⊆ 𝐴) → (𝐹‘(𝑔‘𝑦)) = (𝐴 ∖ (𝑔‘𝑦)))
35	10, 33, 34	syl2an2r 682	. . . . . . . . . 10 ⊢ ((𝑔 ∈ (𝒫 𝐴 ↑m ω) ∧ 𝑦 ∈ ω) → (𝐹‘(𝑔‘𝑦)) = (𝐴 ∖ (𝑔‘𝑦)))
36		peano2 7737	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ω → suc 𝑦 ∈ ω)
37		ffvelrn 6959	. . . . . . . . . . . . 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 4544	. . . . . . . . . . 11 ⊢ ((𝑔 ∈ (𝒫 𝐴 ↑m ω) ∧ 𝑦 ∈ ω) → (𝑔‘suc 𝑦) ⊆ 𝐴)
40	2	isf34lem1 10128	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ V ∧ (𝑔‘suc 𝑦) ⊆ 𝐴) → (𝐹‘(𝑔‘suc 𝑦)) = (𝐴 ∖ (𝑔‘suc 𝑦)))
41	10, 39, 40	syl2an2r 682	. . . . . . . . . 10 ⊢ ((𝑔 ∈ (𝒫 𝐴 ↑m ω) ∧ 𝑦 ∈ ω) → (𝐹‘(𝑔‘suc 𝑦)) = (𝐴 ∖ (𝑔‘suc 𝑦)))
42	35, 41	sseq12d 3954	. . . . . . . . 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	syl5ibr 245	. . . . . . . 8 ⊢ ((𝑔 ∈ (𝒫 𝐴 ↑m ω) ∧ 𝑦 ∈ ω) → ((𝑔‘suc 𝑦) ⊆ (𝑔‘𝑦) → (𝐹‘(𝑔‘𝑦)) ⊆ (𝐹‘(𝑔‘suc 𝑦))))
44		fvco3 6867	. . . . . . . . . 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 6867	. . . . . . . . . 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 3954	. . . . . . . 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 258	. . . . . . 7 ⊢ ((𝑔 ∈ (𝒫 𝐴 ↑m ω) ∧ 𝑦 ∈ ω) → ((𝑔‘suc 𝑦) ⊆ (𝑔‘𝑦) → ((𝐹 ∘ 𝑔)‘𝑦) ⊆ ((𝐹 ∘ 𝑔)‘suc 𝑦)))
50	49	ralimdva 3108	. . . . . 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 6601	. . . . . . . 8 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → 𝐹 Fn 𝒫 𝐴)
52		imassrn 5980	. . . . . . . . 9 ⊢ (𝐹 “ ran 𝑔) ⊆ ran 𝐹
53	12	frnd 6608	. . . . . . . . 9 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → ran 𝐹 ⊆ 𝒫 𝐴)
54	52, 53	sstrid 3932	. . . . . . . 8 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → (𝐹 “ ran 𝑔) ⊆ 𝒫 𝐴)
55		fnfvima 7109	. . . . . . . . 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 1120	. . . . . . . 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 4135	. . . . . . . . . . . . 13 ⊢ (dom 𝐹 ∩ ran 𝑔) = (ran 𝑔 ∩ dom 𝐹)
59	13	frnd 6608	. . . . . . . . . . . . . . 15 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → ran 𝑔 ⊆ 𝒫 𝐴)
60	12	fdmd 6611	. . . . . . . . . . . . . . 15 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → dom 𝐹 = 𝒫 𝐴)
61	59, 60	sseqtrrd 3962	. . . . . . . . . . . . . 14 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → ran 𝑔 ⊆ dom 𝐹)
62		df-ss 3904	. . . . . . . . . . . . . 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 2790	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → (dom 𝐹 ∩ ran 𝑔) = ran 𝑔)
65	13	fdmd 6611	. . . . . . . . . . . . . 14 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → dom 𝑔 = ω)
66		peano1 7735	. . . . . . . . . . . . . . 15 ⊢ ∅ ∈ ω
67		ne0i 4268	. . . . . . . . . . . . . . 15 ⊢ (∅ ∈ ω → ω ≠ ∅)
68	66, 67	mp1i 13	. . . . . . . . . . . . . 14 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → ω ≠ ∅)
69	65, 68	eqnetrd 3011	. . . . . . . . . . . . 13 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → dom 𝑔 ≠ ∅)
70		dm0rn0 5834	. . . . . . . . . . . . . 14 ⊢ (dom 𝑔 = ∅ ↔ ran 𝑔 = ∅)
71	70	necon3bii 2996	. . . . . . . . . . . . 13 ⊢ (dom 𝑔 ≠ ∅ ↔ ran 𝑔 ≠ ∅)
72	69, 71	sylib 217	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → ran 𝑔 ≠ ∅)
73	64, 72	eqnetrd 3011	. . . . . . . . . . 11 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → (dom 𝐹 ∩ ran 𝑔) ≠ ∅)
74		imadisj 5988	. . . . . . . . . . . 12 ⊢ ((𝐹 “ ran 𝑔) = ∅ ↔ (dom 𝐹 ∩ ran 𝑔) = ∅)
75	74	necon3bii 2996	. . . . . . . . . . 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 10133	. . . . . . . . . 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 834	. . . . . . . . 9 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → (𝐹‘∪ (𝐹 “ ran 𝑔)) = ∩ (𝐹 “ (𝐹 “ ran 𝑔)))
79	2	isf34lem3 10131	. . . . . . . . . . 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 4884	. . . . . . . . 9 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → ∩ (𝐹 “ (𝐹 “ ran 𝑔)) = ∩ ran 𝑔)
82	78, 81	eqtrd 2778	. . . . . . . 8 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → (𝐹‘∪ (𝐹 “ ran 𝑔)) = ∩ ran 𝑔)
83	82, 80	eleq12d 2833	. . . . . . 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 3102	. . 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 10124	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinIII ↔ ∀𝑔 ∈ (𝒫 𝐴 ↑m ω)(∀𝑦 ∈ ω (𝑔‘suc 𝑦) ⊆ (𝑔‘𝑦) → ∩ ran 𝑔 ∈ ran 𝑔)))
89	87, 88	syl5ibr 245	. 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 225	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 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  Vcvv 3432   ∖ cdif 3884   ∩ cin 3886   ⊆ wss 3887  ∅c0 4256  𝒫 cpw 4533  ∪ cuni 4839  ∩ cint 4879   ↦ cmpt 5157  dom cdm 5589  ran crn 5590   “ cima 5592   ∘ ccom 5593  suc csuc 6268   Fn wfn 6428  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275  ωcom 7712   ↑_m cmap 8615  Fin^IIIcfin3 10037

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-rpss 7576  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-seqom 8279  df-1o 8297  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-wdom 9324  df-card 9697  df-fin4 10043  df-fin3 10044

This theorem is referenced by:  isfin3-4  10138