Theorem isf34lem6 9802

Description: Lemma for isfin3-4 9804. (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 8428	. . . 4 ⊢ (𝑓 ∈ (𝒫 𝐴 ↑m ω) → 𝑓:ω⟶𝒫 𝐴)
2		compss.a	. . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥))
3	2	isf34lem7 9801	. . . . 5 ⊢ ((𝐴 ∈ FinIII ∧ 𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝑓‘𝑦) ⊆ (𝑓‘suc 𝑦)) → ∪ ran 𝑓 ∈ ran 𝑓)
4	3	3expia 1117	. . . 4 ⊢ ((𝐴 ∈ FinIII ∧ 𝑓:ω⟶𝒫 𝐴) → (∀𝑦 ∈ ω (𝑓‘𝑦) ⊆ (𝑓‘suc 𝑦) → ∪ ran 𝑓 ∈ ran 𝑓))
5	1, 4	sylan2 594	. . 3 ⊢ ((𝐴 ∈ FinIII ∧ 𝑓 ∈ (𝒫 𝐴 ↑m ω)) → (∀𝑦 ∈ ω (𝑓‘𝑦) ⊆ (𝑓‘suc 𝑦) → ∪ ran 𝑓 ∈ ran 𝑓))
6	5	ralrimiva 3182	. 2 ⊢ (𝐴 ∈ FinIII → ∀𝑓 ∈ (𝒫 𝐴 ↑m ω)(∀𝑦 ∈ ω (𝑓‘𝑦) ⊆ (𝑓‘suc 𝑦) → ∪ ran 𝑓 ∈ ran 𝑓))
7		elmapex 8427	. . . . . . . . . . 11 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → (𝒫 𝐴 ∈ V ∧ ω ∈ V))
8	7	simpld 497	. . . . . . . . . 10 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → 𝒫 𝐴 ∈ V)
9		pwexb 7488	. . . . . . . . . 10 ⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
10	8, 9	sylibr 236	. . . . . . . . 9 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → 𝐴 ∈ V)
11	2	isf34lem2 9795	. . . . . . . . 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 8428	. . . . . . . 8 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → 𝑔:ω⟶𝒫 𝐴)
14		fco 6531	. . . . . . . 8 ⊢ ((𝐹:𝒫 𝐴⟶𝒫 𝐴 ∧ 𝑔:ω⟶𝒫 𝐴) → (𝐹 ∘ 𝑔):ω⟶𝒫 𝐴)
15	12, 13, 14	syl2anc 586	. . . . . . 7 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → (𝐹 ∘ 𝑔):ω⟶𝒫 𝐴)
16		elmapg 8419	. . . . . . . 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 259	. . . . . 6 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → (𝐹 ∘ 𝑔) ∈ (𝒫 𝐴 ↑m ω))
19		fveq1 6669	. . . . . . . . . 10 ⊢ (𝑓 = (𝐹 ∘ 𝑔) → (𝑓‘𝑦) = ((𝐹 ∘ 𝑔)‘𝑦))
20		fveq1 6669	. . . . . . . . . 10 ⊢ (𝑓 = (𝐹 ∘ 𝑔) → (𝑓‘suc 𝑦) = ((𝐹 ∘ 𝑔)‘suc 𝑦))
21	19, 20	sseq12d 4000	. . . . . . . . 9 ⊢ (𝑓 = (𝐹 ∘ 𝑔) → ((𝑓‘𝑦) ⊆ (𝑓‘suc 𝑦) ↔ ((𝐹 ∘ 𝑔)‘𝑦) ⊆ ((𝐹 ∘ 𝑔)‘suc 𝑦)))
22	21	ralbidv 3197	. . . . . . . 8 ⊢ (𝑓 = (𝐹 ∘ 𝑔) → (∀𝑦 ∈ ω (𝑓‘𝑦) ⊆ (𝑓‘suc 𝑦) ↔ ∀𝑦 ∈ ω ((𝐹 ∘ 𝑔)‘𝑦) ⊆ ((𝐹 ∘ 𝑔)‘suc 𝑦)))
23		rneq 5806	. . . . . . . . . . 11 ⊢ (𝑓 = (𝐹 ∘ 𝑔) → ran 𝑓 = ran (𝐹 ∘ 𝑔))
24		rnco2 6106	. . . . . . . . . . 11 ⊢ ran (𝐹 ∘ 𝑔) = (𝐹 “ ran 𝑔)
25	23, 24	syl6eq 2872	. . . . . . . . . 10 ⊢ (𝑓 = (𝐹 ∘ 𝑔) → ran 𝑓 = (𝐹 “ ran 𝑔))
26	25	unieqd 4852	. . . . . . . . 9 ⊢ (𝑓 = (𝐹 ∘ 𝑔) → ∪ ran 𝑓 = ∪ (𝐹 “ ran 𝑔))
27	26, 25	eleq12d 2907	. . . . . . . 8 ⊢ (𝑓 = (𝐹 ∘ 𝑔) → (∪ ran 𝑓 ∈ ran 𝑓 ↔ ∪ (𝐹 “ ran 𝑔) ∈ (𝐹 “ ran 𝑔)))
28	22, 27	imbi12d 347	. . . . . . 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 3620	. . . . . 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 4115	. . . . . . . . 9 ⊢ ((𝑔‘suc 𝑦) ⊆ (𝑔‘𝑦) → (𝐴 ∖ (𝑔‘𝑦)) ⊆ (𝐴 ∖ (𝑔‘suc 𝑦)))
32	13	ffvelrnda 6851	. . . . . . . . . . . 12 ⊢ ((𝑔 ∈ (𝒫 𝐴 ↑m ω) ∧ 𝑦 ∈ ω) → (𝑔‘𝑦) ∈ 𝒫 𝐴)
33	32	elpwid 4550	. . . . . . . . . . 11 ⊢ ((𝑔 ∈ (𝒫 𝐴 ↑m ω) ∧ 𝑦 ∈ ω) → (𝑔‘𝑦) ⊆ 𝐴)
34	2	isf34lem1 9794	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ V ∧ (𝑔‘𝑦) ⊆ 𝐴) → (𝐹‘(𝑔‘𝑦)) = (𝐴 ∖ (𝑔‘𝑦)))
35	10, 33, 34	syl2an2r 683	. . . . . . . . . 10 ⊢ ((𝑔 ∈ (𝒫 𝐴 ↑m ω) ∧ 𝑦 ∈ ω) → (𝐹‘(𝑔‘𝑦)) = (𝐴 ∖ (𝑔‘𝑦)))
36		peano2 7602	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ω → suc 𝑦 ∈ ω)
37		ffvelrn 6849	. . . . . . . . . . . . 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 4550	. . . . . . . . . . 11 ⊢ ((𝑔 ∈ (𝒫 𝐴 ↑m ω) ∧ 𝑦 ∈ ω) → (𝑔‘suc 𝑦) ⊆ 𝐴)
40	2	isf34lem1 9794	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ V ∧ (𝑔‘suc 𝑦) ⊆ 𝐴) → (𝐹‘(𝑔‘suc 𝑦)) = (𝐴 ∖ (𝑔‘suc 𝑦)))
41	10, 39, 40	syl2an2r 683	. . . . . . . . . 10 ⊢ ((𝑔 ∈ (𝒫 𝐴 ↑m ω) ∧ 𝑦 ∈ ω) → (𝐹‘(𝑔‘suc 𝑦)) = (𝐴 ∖ (𝑔‘suc 𝑦)))
42	35, 41	sseq12d 4000	. . . . . . . . 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 248	. . . . . . . 8 ⊢ ((𝑔 ∈ (𝒫 𝐴 ↑m ω) ∧ 𝑦 ∈ ω) → ((𝑔‘suc 𝑦) ⊆ (𝑔‘𝑦) → (𝐹‘(𝑔‘𝑦)) ⊆ (𝐹‘(𝑔‘suc 𝑦))))
44		fvco3 6760	. . . . . . . . . 10 ⊢ ((𝑔:ω⟶𝒫 𝐴 ∧ 𝑦 ∈ ω) → ((𝐹 ∘ 𝑔)‘𝑦) = (𝐹‘(𝑔‘𝑦)))
45	13, 44	sylan 582	. . . . . . . . 9 ⊢ ((𝑔 ∈ (𝒫 𝐴 ↑m ω) ∧ 𝑦 ∈ ω) → ((𝐹 ∘ 𝑔)‘𝑦) = (𝐹‘(𝑔‘𝑦)))
46		fvco3 6760	. . . . . . . . . 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 4000	. . . . . . . 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 261	. . . . . . 7 ⊢ ((𝑔 ∈ (𝒫 𝐴 ↑m ω) ∧ 𝑦 ∈ ω) → ((𝑔‘suc 𝑦) ⊆ (𝑔‘𝑦) → ((𝐹 ∘ 𝑔)‘𝑦) ⊆ ((𝐹 ∘ 𝑔)‘suc 𝑦)))
50	49	ralimdva 3177	. . . . . 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 6515	. . . . . . . 8 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → 𝐹 Fn 𝒫 𝐴)
52		imassrn 5940	. . . . . . . . 9 ⊢ (𝐹 “ ran 𝑔) ⊆ ran 𝐹
53	12	frnd 6521	. . . . . . . . 9 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → ran 𝐹 ⊆ 𝒫 𝐴)
54	52, 53	sstrid 3978	. . . . . . . 8 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → (𝐹 “ ran 𝑔) ⊆ 𝒫 𝐴)
55		fnfvima 6995	. . . . . . . . 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 1117	. . . . . . . 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 586	. . . . . . 7 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → (∪ (𝐹 “ ran 𝑔) ∈ (𝐹 “ ran 𝑔) → (𝐹‘∪ (𝐹 “ ran 𝑔)) ∈ (𝐹 “ (𝐹 “ ran 𝑔))))
58		incom 4178	. . . . . . . . . . . . 13 ⊢ (dom 𝐹 ∩ ran 𝑔) = (ran 𝑔 ∩ dom 𝐹)
59	13	frnd 6521	. . . . . . . . . . . . . . 15 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → ran 𝑔 ⊆ 𝒫 𝐴)
60	12	fdmd 6523	. . . . . . . . . . . . . . 15 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → dom 𝐹 = 𝒫 𝐴)
61	59, 60	sseqtrrd 4008	. . . . . . . . . . . . . 14 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → ran 𝑔 ⊆ dom 𝐹)
62		df-ss 3952	. . . . . . . . . . . . . 14 ⊢ (ran 𝑔 ⊆ dom 𝐹 ↔ (ran 𝑔 ∩ dom 𝐹) = ran 𝑔)
63	61, 62	sylib 220	. . . . . . . . . . . . 13 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → (ran 𝑔 ∩ dom 𝐹) = ran 𝑔)
64	58, 63	syl5eq 2868	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → (dom 𝐹 ∩ ran 𝑔) = ran 𝑔)
65	13	fdmd 6523	. . . . . . . . . . . . . 14 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → dom 𝑔 = ω)
66		peano1 7601	. . . . . . . . . . . . . . 15 ⊢ ∅ ∈ ω
67		ne0i 4300	. . . . . . . . . . . . . . 15 ⊢ (∅ ∈ ω → ω ≠ ∅)
68	66, 67	mp1i 13	. . . . . . . . . . . . . 14 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → ω ≠ ∅)
69	65, 68	eqnetrd 3083	. . . . . . . . . . . . 13 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → dom 𝑔 ≠ ∅)
70		dm0rn0 5795	. . . . . . . . . . . . . 14 ⊢ (dom 𝑔 = ∅ ↔ ran 𝑔 = ∅)
71	70	necon3bii 3068	. . . . . . . . . . . . 13 ⊢ (dom 𝑔 ≠ ∅ ↔ ran 𝑔 ≠ ∅)
72	69, 71	sylib 220	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → ran 𝑔 ≠ ∅)
73	64, 72	eqnetrd 3083	. . . . . . . . . . 11 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → (dom 𝐹 ∩ ran 𝑔) ≠ ∅)
74		imadisj 5948	. . . . . . . . . . . 12 ⊢ ((𝐹 “ ran 𝑔) = ∅ ↔ (dom 𝐹 ∩ ran 𝑔) = ∅)
75	74	necon3bii 3068	. . . . . . . . . . 11 ⊢ ((𝐹 “ ran 𝑔) ≠ ∅ ↔ (dom 𝐹 ∩ ran 𝑔) ≠ ∅)
76	73, 75	sylibr 236	. . . . . . . . . 10 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → (𝐹 “ ran 𝑔) ≠ ∅)
77	2	isf34lem4 9799	. . . . . . . . . 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 9797	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ V ∧ ran 𝑔 ⊆ 𝒫 𝐴) → (𝐹 “ (𝐹 “ ran 𝑔)) = ran 𝑔)
80	10, 59, 79	syl2anc 586	. . . . . . . . . 10 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → (𝐹 “ (𝐹 “ ran 𝑔)) = ran 𝑔)
81	80	inteqd 4881	. . . . . . . . 9 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → ∩ (𝐹 “ (𝐹 “ ran 𝑔)) = ∩ ran 𝑔)
82	78, 81	eqtrd 2856	. . . . . . . 8 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → (𝐹‘∪ (𝐹 “ ran 𝑔)) = ∩ ran 𝑔)
83	82, 80	eleq12d 2907	. . . . . . 7 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑m ω) → ((𝐹‘∪ (𝐹 “ ran 𝑔)) ∈ (𝐹 “ (𝐹 “ ran 𝑔)) ↔ ∩ ran 𝑔 ∈ ran 𝑔))
84	57, 83	sylibd 241	. . . . . 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 3181	. . 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 9790	. . 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 248	. 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 228	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 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114   ≠ wne 3016  ∀wral 3138  Vcvv 3494   ∖ cdif 3933   ∩ cin 3935   ⊆ wss 3936  ∅c0 4291  𝒫 cpw 4539  ∪ cuni 4838  ∩ cint 4876   ↦ cmpt 5146  dom cdm 5555  ran crn 5556   “ cima 5558   ∘ ccom 5559  suc csuc 6193   Fn wfn 6350  ⟶wf 6351  ‘cfv 6355  (class class class)co 7156  ωcom 7580   ↑_m cmap 8406  Fin^IIIcfin3 9703

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-se 5515  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-isom 6364  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-rpss 7449  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-seqom 8084  df-1o 8102  df-oadd 8106  df-er 8289  df-map 8408  df-en 8510  df-dom 8511  df-sdom 8512  df-fin 8513  df-wdom 9023  df-card 9368  df-fin4 9709  df-fin3 9710

This theorem is referenced by:  isfin3-4  9804