Theorem isf34lem6 9202

Description: Lemma for isfin3-4 9204. (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 7879	. . . 4 ⊢ (𝑓 ∈ (𝒫 𝐴 ↑𝑚 ω) → 𝑓:ω⟶𝒫 𝐴)
2		compss.a	. . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥))
3	2	isf34lem7 9201	. . . . 5 ⊢ ((𝐴 ∈ FinIII ∧ 𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝑓‘𝑦) ⊆ (𝑓‘suc 𝑦)) → ∪ ran 𝑓 ∈ ran 𝑓)
4	3	3expia 1267	. . . 4 ⊢ ((𝐴 ∈ FinIII ∧ 𝑓:ω⟶𝒫 𝐴) → (∀𝑦 ∈ ω (𝑓‘𝑦) ⊆ (𝑓‘suc 𝑦) → ∪ ran 𝑓 ∈ ran 𝑓))
5	1, 4	sylan2 491	. . 3 ⊢ ((𝐴 ∈ FinIII ∧ 𝑓 ∈ (𝒫 𝐴 ↑𝑚 ω)) → (∀𝑦 ∈ ω (𝑓‘𝑦) ⊆ (𝑓‘suc 𝑦) → ∪ ran 𝑓 ∈ ran 𝑓))
6	5	ralrimiva 2966	. 2 ⊢ (𝐴 ∈ FinIII → ∀𝑓 ∈ (𝒫 𝐴 ↑𝑚 ω)(∀𝑦 ∈ ω (𝑓‘𝑦) ⊆ (𝑓‘suc 𝑦) → ∪ ran 𝑓 ∈ ran 𝑓))
7		elmapex 7878	. . . . . . . . . . 11 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑𝑚 ω) → (𝒫 𝐴 ∈ V ∧ ω ∈ V))
8	7	simpld 475	. . . . . . . . . 10 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑𝑚 ω) → 𝒫 𝐴 ∈ V)
9		pwexb 6975	. . . . . . . . . 10 ⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
10	8, 9	sylibr 224	. . . . . . . . 9 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑𝑚 ω) → 𝐴 ∈ V)
11	2	isf34lem2 9195	. . . . . . . . 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 7879	. . . . . . . 8 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑𝑚 ω) → 𝑔:ω⟶𝒫 𝐴)
14		fco 6058	. . . . . . . 8 ⊢ ((𝐹:𝒫 𝐴⟶𝒫 𝐴 ∧ 𝑔:ω⟶𝒫 𝐴) → (𝐹 ∘ 𝑔):ω⟶𝒫 𝐴)
15	12, 13, 14	syl2anc 693	. . . . . . 7 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑𝑚 ω) → (𝐹 ∘ 𝑔):ω⟶𝒫 𝐴)
16		elmapg 7870	. . . . . . . 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 247	. . . . . 6 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑𝑚 ω) → (𝐹 ∘ 𝑔) ∈ (𝒫 𝐴 ↑𝑚 ω))
19		fveq1 6190	. . . . . . . . . 10 ⊢ (𝑓 = (𝐹 ∘ 𝑔) → (𝑓‘𝑦) = ((𝐹 ∘ 𝑔)‘𝑦))
20		fveq1 6190	. . . . . . . . . 10 ⊢ (𝑓 = (𝐹 ∘ 𝑔) → (𝑓‘suc 𝑦) = ((𝐹 ∘ 𝑔)‘suc 𝑦))
21	19, 20	sseq12d 3634	. . . . . . . . 9 ⊢ (𝑓 = (𝐹 ∘ 𝑔) → ((𝑓‘𝑦) ⊆ (𝑓‘suc 𝑦) ↔ ((𝐹 ∘ 𝑔)‘𝑦) ⊆ ((𝐹 ∘ 𝑔)‘suc 𝑦)))
22	21	ralbidv 2986	. . . . . . . 8 ⊢ (𝑓 = (𝐹 ∘ 𝑔) → (∀𝑦 ∈ ω (𝑓‘𝑦) ⊆ (𝑓‘suc 𝑦) ↔ ∀𝑦 ∈ ω ((𝐹 ∘ 𝑔)‘𝑦) ⊆ ((𝐹 ∘ 𝑔)‘suc 𝑦)))
23		rneq 5351	. . . . . . . . . . 11 ⊢ (𝑓 = (𝐹 ∘ 𝑔) → ran 𝑓 = ran (𝐹 ∘ 𝑔))
24		rnco2 5642	. . . . . . . . . . 11 ⊢ ran (𝐹 ∘ 𝑔) = (𝐹 “ ran 𝑔)
25	23, 24	syl6eq 2672	. . . . . . . . . 10 ⊢ (𝑓 = (𝐹 ∘ 𝑔) → ran 𝑓 = (𝐹 “ ran 𝑔))
26	25	unieqd 4446	. . . . . . . . 9 ⊢ (𝑓 = (𝐹 ∘ 𝑔) → ∪ ran 𝑓 = ∪ (𝐹 “ ran 𝑔))
27	26, 25	eleq12d 2695	. . . . . . . 8 ⊢ (𝑓 = (𝐹 ∘ 𝑔) → (∪ ran 𝑓 ∈ ran 𝑓 ↔ ∪ (𝐹 “ ran 𝑔) ∈ (𝐹 “ ran 𝑔)))
28	22, 27	imbi12d 334	. . . . . . 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 3306	. . . . . 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 3744	. . . . . . . . 9 ⊢ ((𝑔‘suc 𝑦) ⊆ (𝑔‘𝑦) → (𝐴 ∖ (𝑔‘𝑦)) ⊆ (𝐴 ∖ (𝑔‘suc 𝑦)))
32	10	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑔 ∈ (𝒫 𝐴 ↑𝑚 ω) ∧ 𝑦 ∈ ω) → 𝐴 ∈ V)
33	13	ffvelrnda 6359	. . . . . . . . . . . 12 ⊢ ((𝑔 ∈ (𝒫 𝐴 ↑𝑚 ω) ∧ 𝑦 ∈ ω) → (𝑔‘𝑦) ∈ 𝒫 𝐴)
34	33	elpwid 4170	. . . . . . . . . . 11 ⊢ ((𝑔 ∈ (𝒫 𝐴 ↑𝑚 ω) ∧ 𝑦 ∈ ω) → (𝑔‘𝑦) ⊆ 𝐴)
35	2	isf34lem1 9194	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ V ∧ (𝑔‘𝑦) ⊆ 𝐴) → (𝐹‘(𝑔‘𝑦)) = (𝐴 ∖ (𝑔‘𝑦)))
36	32, 34, 35	syl2anc 693	. . . . . . . . . 10 ⊢ ((𝑔 ∈ (𝒫 𝐴 ↑𝑚 ω) ∧ 𝑦 ∈ ω) → (𝐹‘(𝑔‘𝑦)) = (𝐴 ∖ (𝑔‘𝑦)))
37		peano2 7086	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ω → suc 𝑦 ∈ ω)
38		ffvelrn 6357	. . . . . . . . . . . . 13 ⊢ ((𝑔:ω⟶𝒫 𝐴 ∧ suc 𝑦 ∈ ω) → (𝑔‘suc 𝑦) ∈ 𝒫 𝐴)
39	13, 37, 38	syl2an 494	. . . . . . . . . . . 12 ⊢ ((𝑔 ∈ (𝒫 𝐴 ↑𝑚 ω) ∧ 𝑦 ∈ ω) → (𝑔‘suc 𝑦) ∈ 𝒫 𝐴)
40	39	elpwid 4170	. . . . . . . . . . 11 ⊢ ((𝑔 ∈ (𝒫 𝐴 ↑𝑚 ω) ∧ 𝑦 ∈ ω) → (𝑔‘suc 𝑦) ⊆ 𝐴)
41	2	isf34lem1 9194	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ V ∧ (𝑔‘suc 𝑦) ⊆ 𝐴) → (𝐹‘(𝑔‘suc 𝑦)) = (𝐴 ∖ (𝑔‘suc 𝑦)))
42	32, 40, 41	syl2anc 693	. . . . . . . . . 10 ⊢ ((𝑔 ∈ (𝒫 𝐴 ↑𝑚 ω) ∧ 𝑦 ∈ ω) → (𝐹‘(𝑔‘suc 𝑦)) = (𝐴 ∖ (𝑔‘suc 𝑦)))
43	36, 42	sseq12d 3634	. . . . . . . . 9 ⊢ ((𝑔 ∈ (𝒫 𝐴 ↑𝑚 ω) ∧ 𝑦 ∈ ω) → ((𝐹‘(𝑔‘𝑦)) ⊆ (𝐹‘(𝑔‘suc 𝑦)) ↔ (𝐴 ∖ (𝑔‘𝑦)) ⊆ (𝐴 ∖ (𝑔‘suc 𝑦))))
44	31, 43	syl5ibr 236	. . . . . . . 8 ⊢ ((𝑔 ∈ (𝒫 𝐴 ↑𝑚 ω) ∧ 𝑦 ∈ ω) → ((𝑔‘suc 𝑦) ⊆ (𝑔‘𝑦) → (𝐹‘(𝑔‘𝑦)) ⊆ (𝐹‘(𝑔‘suc 𝑦))))
45		fvco3 6275	. . . . . . . . . 10 ⊢ ((𝑔:ω⟶𝒫 𝐴 ∧ 𝑦 ∈ ω) → ((𝐹 ∘ 𝑔)‘𝑦) = (𝐹‘(𝑔‘𝑦)))
46	13, 45	sylan 488	. . . . . . . . 9 ⊢ ((𝑔 ∈ (𝒫 𝐴 ↑𝑚 ω) ∧ 𝑦 ∈ ω) → ((𝐹 ∘ 𝑔)‘𝑦) = (𝐹‘(𝑔‘𝑦)))
47		fvco3 6275	. . . . . . . . . 10 ⊢ ((𝑔:ω⟶𝒫 𝐴 ∧ suc 𝑦 ∈ ω) → ((𝐹 ∘ 𝑔)‘suc 𝑦) = (𝐹‘(𝑔‘suc 𝑦)))
48	13, 37, 47	syl2an 494	. . . . . . . . 9 ⊢ ((𝑔 ∈ (𝒫 𝐴 ↑𝑚 ω) ∧ 𝑦 ∈ ω) → ((𝐹 ∘ 𝑔)‘suc 𝑦) = (𝐹‘(𝑔‘suc 𝑦)))
49	46, 48	sseq12d 3634	. . . . . . . 8 ⊢ ((𝑔 ∈ (𝒫 𝐴 ↑𝑚 ω) ∧ 𝑦 ∈ ω) → (((𝐹 ∘ 𝑔)‘𝑦) ⊆ ((𝐹 ∘ 𝑔)‘suc 𝑦) ↔ (𝐹‘(𝑔‘𝑦)) ⊆ (𝐹‘(𝑔‘suc 𝑦))))
50	44, 49	sylibrd 249	. . . . . . 7 ⊢ ((𝑔 ∈ (𝒫 𝐴 ↑𝑚 ω) ∧ 𝑦 ∈ ω) → ((𝑔‘suc 𝑦) ⊆ (𝑔‘𝑦) → ((𝐹 ∘ 𝑔)‘𝑦) ⊆ ((𝐹 ∘ 𝑔)‘suc 𝑦)))
51	50	ralimdva 2962	. . . . . 6 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑𝑚 ω) → (∀𝑦 ∈ ω (𝑔‘suc 𝑦) ⊆ (𝑔‘𝑦) → ∀𝑦 ∈ ω ((𝐹 ∘ 𝑔)‘𝑦) ⊆ ((𝐹 ∘ 𝑔)‘suc 𝑦)))
52		ffn 6045	. . . . . . . . 9 ⊢ (𝐹:𝒫 𝐴⟶𝒫 𝐴 → 𝐹 Fn 𝒫 𝐴)
53	12, 52	syl 17	. . . . . . . 8 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑𝑚 ω) → 𝐹 Fn 𝒫 𝐴)
54		imassrn 5477	. . . . . . . . 9 ⊢ (𝐹 “ ran 𝑔) ⊆ ran 𝐹
55		frn 6053	. . . . . . . . . 10 ⊢ (𝐹:𝒫 𝐴⟶𝒫 𝐴 → ran 𝐹 ⊆ 𝒫 𝐴)
56	12, 55	syl 17	. . . . . . . . 9 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑𝑚 ω) → ran 𝐹 ⊆ 𝒫 𝐴)
57	54, 56	syl5ss 3614	. . . . . . . 8 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑𝑚 ω) → (𝐹 “ ran 𝑔) ⊆ 𝒫 𝐴)
58		fnfvima 6496	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝒫 𝐴 ∧ (𝐹 “ ran 𝑔) ⊆ 𝒫 𝐴 ∧ ∪ (𝐹 “ ran 𝑔) ∈ (𝐹 “ ran 𝑔)) → (𝐹‘∪ (𝐹 “ ran 𝑔)) ∈ (𝐹 “ (𝐹 “ ran 𝑔)))
59	58	3expia 1267	. . . . . . . 8 ⊢ ((𝐹 Fn 𝒫 𝐴 ∧ (𝐹 “ ran 𝑔) ⊆ 𝒫 𝐴) → (∪ (𝐹 “ ran 𝑔) ∈ (𝐹 “ ran 𝑔) → (𝐹‘∪ (𝐹 “ ran 𝑔)) ∈ (𝐹 “ (𝐹 “ ran 𝑔))))
60	53, 57, 59	syl2anc 693	. . . . . . 7 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑𝑚 ω) → (∪ (𝐹 “ ran 𝑔) ∈ (𝐹 “ ran 𝑔) → (𝐹‘∪ (𝐹 “ ran 𝑔)) ∈ (𝐹 “ (𝐹 “ ran 𝑔))))
61		incom 3805	. . . . . . . . . . . . 13 ⊢ (dom 𝐹 ∩ ran 𝑔) = (ran 𝑔 ∩ dom 𝐹)
62		frn 6053	. . . . . . . . . . . . . . . 16 ⊢ (𝑔:ω⟶𝒫 𝐴 → ran 𝑔 ⊆ 𝒫 𝐴)
63	13, 62	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑𝑚 ω) → ran 𝑔 ⊆ 𝒫 𝐴)
64		fdm 6051	. . . . . . . . . . . . . . . 16 ⊢ (𝐹:𝒫 𝐴⟶𝒫 𝐴 → dom 𝐹 = 𝒫 𝐴)
65	12, 64	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑𝑚 ω) → dom 𝐹 = 𝒫 𝐴)
66	63, 65	sseqtr4d 3642	. . . . . . . . . . . . . 14 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑𝑚 ω) → ran 𝑔 ⊆ dom 𝐹)
67		df-ss 3588	. . . . . . . . . . . . . 14 ⊢ (ran 𝑔 ⊆ dom 𝐹 ↔ (ran 𝑔 ∩ dom 𝐹) = ran 𝑔)
68	66, 67	sylib 208	. . . . . . . . . . . . 13 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑𝑚 ω) → (ran 𝑔 ∩ dom 𝐹) = ran 𝑔)
69	61, 68	syl5eq 2668	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑𝑚 ω) → (dom 𝐹 ∩ ran 𝑔) = ran 𝑔)
70		fdm 6051	. . . . . . . . . . . . . . 15 ⊢ (𝑔:ω⟶𝒫 𝐴 → dom 𝑔 = ω)
71	13, 70	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑𝑚 ω) → dom 𝑔 = ω)
72		peano1 7085	. . . . . . . . . . . . . . 15 ⊢ ∅ ∈ ω
73		ne0i 3921	. . . . . . . . . . . . . . 15 ⊢ (∅ ∈ ω → ω ≠ ∅)
74	72, 73	mp1i 13	. . . . . . . . . . . . . 14 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑𝑚 ω) → ω ≠ ∅)
75	71, 74	eqnetrd 2861	. . . . . . . . . . . . 13 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑𝑚 ω) → dom 𝑔 ≠ ∅)
76		dm0rn0 5342	. . . . . . . . . . . . . 14 ⊢ (dom 𝑔 = ∅ ↔ ran 𝑔 = ∅)
77	76	necon3bii 2846	. . . . . . . . . . . . 13 ⊢ (dom 𝑔 ≠ ∅ ↔ ran 𝑔 ≠ ∅)
78	75, 77	sylib 208	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑𝑚 ω) → ran 𝑔 ≠ ∅)
79	69, 78	eqnetrd 2861	. . . . . . . . . . 11 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑𝑚 ω) → (dom 𝐹 ∩ ran 𝑔) ≠ ∅)
80		imadisj 5484	. . . . . . . . . . . 12 ⊢ ((𝐹 “ ran 𝑔) = ∅ ↔ (dom 𝐹 ∩ ran 𝑔) = ∅)
81	80	necon3bii 2846	. . . . . . . . . . 11 ⊢ ((𝐹 “ ran 𝑔) ≠ ∅ ↔ (dom 𝐹 ∩ ran 𝑔) ≠ ∅)
82	79, 81	sylibr 224	. . . . . . . . . 10 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑𝑚 ω) → (𝐹 “ ran 𝑔) ≠ ∅)
83	2	isf34lem4 9199	. . . . . . . . . 10 ⊢ ((𝐴 ∈ V ∧ ((𝐹 “ ran 𝑔) ⊆ 𝒫 𝐴 ∧ (𝐹 “ ran 𝑔) ≠ ∅)) → (𝐹‘∪ (𝐹 “ ran 𝑔)) = ∩ (𝐹 “ (𝐹 “ ran 𝑔)))
84	10, 57, 82, 83	syl12anc 1324	. . . . . . . . 9 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑𝑚 ω) → (𝐹‘∪ (𝐹 “ ran 𝑔)) = ∩ (𝐹 “ (𝐹 “ ran 𝑔)))
85	2	isf34lem3 9197	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ V ∧ ran 𝑔 ⊆ 𝒫 𝐴) → (𝐹 “ (𝐹 “ ran 𝑔)) = ran 𝑔)
86	10, 63, 85	syl2anc 693	. . . . . . . . . 10 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑𝑚 ω) → (𝐹 “ (𝐹 “ ran 𝑔)) = ran 𝑔)
87	86	inteqd 4480	. . . . . . . . 9 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑𝑚 ω) → ∩ (𝐹 “ (𝐹 “ ran 𝑔)) = ∩ ran 𝑔)
88	84, 87	eqtrd 2656	. . . . . . . 8 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑𝑚 ω) → (𝐹‘∪ (𝐹 “ ran 𝑔)) = ∩ ran 𝑔)
89	88, 86	eleq12d 2695	. . . . . . 7 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑𝑚 ω) → ((𝐹‘∪ (𝐹 “ ran 𝑔)) ∈ (𝐹 “ (𝐹 “ ran 𝑔)) ↔ ∩ ran 𝑔 ∈ ran 𝑔))
90	60, 89	sylibd 229	. . . . . 6 ⊢ (𝑔 ∈ (𝒫 𝐴 ↑𝑚 ω) → (∪ (𝐹 “ ran 𝑔) ∈ (𝐹 “ ran 𝑔) → ∩ ran 𝑔 ∈ ran 𝑔))
91	51, 90	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)))
92	30, 91	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)))
93	92	ralrimiv 2965	. . 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))
94		isfin3-3 9190	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinIII ↔ ∀𝑔 ∈ (𝒫 𝐴 ↑𝑚 ω)(∀𝑦 ∈ ω (𝑔‘suc 𝑦) ⊆ (𝑔‘𝑦) → ∩ ran 𝑔 ∈ ran 𝑔)))
95	93, 94	syl5ibr 236	. 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑓 ∈ (𝒫 𝐴 ↑𝑚 ω)(∀𝑦 ∈ ω (𝑓‘𝑦) ⊆ (𝑓‘suc 𝑦) → ∪ ran 𝑓 ∈ ran 𝑓) → 𝐴 ∈ FinIII))
96	6, 95	impbid2 216	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 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  Vcvv 3200   ∖ cdif 3571   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  𝒫 cpw 4158  ∪ cuni 4436  ∩ cint 4475   ↦ cmpt 4729  dom cdm 5114  ran crn 5115   “ cima 5117   ∘ ccom 5118  suc csuc 5725   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  ωcom 7065   ↑_𝑚 cmap 7857  Fin^IIIcfin3 9103

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-rpss 6937  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-seqom 7543  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-wdom 8464  df-card 8765  df-fin4 9109  df-fin3 9110

This theorem is referenced by:  isfin3-4  9204