Theorem fin23lem30 9164

Description: Lemma for fin23 9211. The residual is disjoint from the common set. (Contributed by Stefan O'Rear, 2-Nov-2014.)

Hypotheses

Ref	Expression
fin23lem.a	⊢ 𝑈 = seq𝜔((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢))), ∪ ran 𝑡)
fin23lem17.f	⊢ 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑𝑚 ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)}
fin23lem.b	⊢ 𝑃 = {𝑣 ∈ ω ∣ ∩ ran 𝑈 ⊆ (𝑡‘𝑣)}
fin23lem.c	⊢ 𝑄 = (𝑤 ∈ ω ↦ (℩𝑥 ∈ 𝑃 (𝑥 ∩ 𝑃) ≈ 𝑤))
fin23lem.d	⊢ 𝑅 = (𝑤 ∈ ω ↦ (℩𝑥 ∈ (ω ∖ 𝑃)(𝑥 ∩ (ω ∖ 𝑃)) ≈ 𝑤))
fin23lem.e	⊢ 𝑍 = if(𝑃 ∈ Fin, (𝑡 ∘ 𝑅), ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄))

Assertion

Ref	Expression
fin23lem30	⊢ (Fun 𝑡 → (∪ ran 𝑍 ∩ ∩ ran 𝑈) = ∅)

Distinct variable groups:   𝑔,𝑖,𝑡,𝑢,𝑣,𝑥,𝑧,𝑎   𝐹,𝑎,𝑡   𝑤,𝑎,𝑥,𝑧,𝑃   𝑣,𝑎,𝑅,𝑖,𝑢   𝑈,𝑎,𝑖,𝑢,𝑣,𝑧   𝑍,𝑎   𝑔,𝑎

Allowed substitution hints:   𝑃(𝑣,𝑢,𝑡,𝑔,𝑖)   𝑄(𝑥,𝑧,𝑤,𝑣,𝑢,𝑡,𝑔,𝑖,𝑎)   𝑅(𝑥,𝑧,𝑤,𝑡,𝑔)   𝑈(𝑥,𝑤,𝑡,𝑔)   𝐹(𝑥,𝑧,𝑤,𝑣,𝑢,𝑔,𝑖)   𝑍(𝑥,𝑧,𝑤,𝑣,𝑢,𝑡,𝑔,𝑖)

Proof of Theorem fin23lem30

Dummy variable 𝑏 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fin23lem.e	. 2 ⊢ 𝑍 = if(𝑃 ∈ Fin, (𝑡 ∘ 𝑅), ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄))
2		eqif 4126	. . 3 ⊢ (𝑍 = if(𝑃 ∈ Fin, (𝑡 ∘ 𝑅), ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄)) ↔ ((𝑃 ∈ Fin ∧ 𝑍 = (𝑡 ∘ 𝑅)) ∨ (¬ 𝑃 ∈ Fin ∧ 𝑍 = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄))))
3	2	biimpi 206	. 2 ⊢ (𝑍 = if(𝑃 ∈ Fin, (𝑡 ∘ 𝑅), ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄)) → ((𝑃 ∈ Fin ∧ 𝑍 = (𝑡 ∘ 𝑅)) ∨ (¬ 𝑃 ∈ Fin ∧ 𝑍 = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄))))
4		simpr 477	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ Fin ∧ Fun 𝑡) → Fun 𝑡)
5		fin23lem.d	. . . . . . . . . . . 12 ⊢ 𝑅 = (𝑤 ∈ ω ↦ (℩𝑥 ∈ (ω ∖ 𝑃)(𝑥 ∩ (ω ∖ 𝑃)) ≈ 𝑤))
6	5	funmpt2 5927	. . . . . . . . . . 11 ⊢ Fun 𝑅
7		funco 5928	. . . . . . . . . . 11 ⊢ ((Fun 𝑡 ∧ Fun 𝑅) → Fun (𝑡 ∘ 𝑅))
8	4, 6, 7	sylancl 694	. . . . . . . . . 10 ⊢ ((𝑃 ∈ Fin ∧ Fun 𝑡) → Fun (𝑡 ∘ 𝑅))
9		elunirn 6509	. . . . . . . . . 10 ⊢ (Fun (𝑡 ∘ 𝑅) → (𝑎 ∈ ∪ ran (𝑡 ∘ 𝑅) ↔ ∃𝑏 ∈ dom (𝑡 ∘ 𝑅)𝑎 ∈ ((𝑡 ∘ 𝑅)‘𝑏)))
10	8, 9	syl 17	. . . . . . . . 9 ⊢ ((𝑃 ∈ Fin ∧ Fun 𝑡) → (𝑎 ∈ ∪ ran (𝑡 ∘ 𝑅) ↔ ∃𝑏 ∈ dom (𝑡 ∘ 𝑅)𝑎 ∈ ((𝑡 ∘ 𝑅)‘𝑏)))
11		dmcoss 5385	. . . . . . . . . . . 12 ⊢ dom (𝑡 ∘ 𝑅) ⊆ dom 𝑅
12	11	sseli 3599	. . . . . . . . . . 11 ⊢ (𝑏 ∈ dom (𝑡 ∘ 𝑅) → 𝑏 ∈ dom 𝑅)
13		fvco 6274	. . . . . . . . . . . . . . . 16 ⊢ ((Fun 𝑅 ∧ 𝑏 ∈ dom 𝑅) → ((𝑡 ∘ 𝑅)‘𝑏) = (𝑡‘(𝑅‘𝑏)))
14	6, 13	mpan 706	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ∈ dom 𝑅 → ((𝑡 ∘ 𝑅)‘𝑏) = (𝑡‘(𝑅‘𝑏)))
15	14	adantl 482	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ((𝑡 ∘ 𝑅)‘𝑏) = (𝑡‘(𝑅‘𝑏)))
16	15	eleq2d 2687	. . . . . . . . . . . . 13 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → (𝑎 ∈ ((𝑡 ∘ 𝑅)‘𝑏) ↔ 𝑎 ∈ (𝑡‘(𝑅‘𝑏))))
17		incom 3805	. . . . . . . . . . . . . . . 16 ⊢ ((𝑡‘(𝑅‘𝑏)) ∩ ∩ ran 𝑈) = (∩ ran 𝑈 ∩ (𝑡‘(𝑅‘𝑏)))
18		difss 3737	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ω ∖ 𝑃) ⊆ ω
19		ominf 8172	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ¬ ω ∈ Fin
20		fin23lem.b	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ 𝑃 = {𝑣 ∈ ω ∣ ∩ ran 𝑈 ⊆ (𝑡‘𝑣)}
21		ssrab2 3687	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ {𝑣 ∈ ω ∣ ∩ ran 𝑈 ⊆ (𝑡‘𝑣)} ⊆ ω
22	20, 21	eqsstri 3635	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 𝑃 ⊆ ω
23		undif 4049	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑃 ⊆ ω ↔ (𝑃 ∪ (ω ∖ 𝑃)) = ω)
24	22, 23	mpbi 220	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑃 ∪ (ω ∖ 𝑃)) = ω
25		unfi 8227	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑃 ∈ Fin ∧ (ω ∖ 𝑃) ∈ Fin) → (𝑃 ∪ (ω ∖ 𝑃)) ∈ Fin)
26	24, 25	syl5eqelr 2706	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑃 ∈ Fin ∧ (ω ∖ 𝑃) ∈ Fin) → ω ∈ Fin)
27	26	ex 450	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑃 ∈ Fin → ((ω ∖ 𝑃) ∈ Fin → ω ∈ Fin))
28	19, 27	mtoi 190	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑃 ∈ Fin → ¬ (ω ∖ 𝑃) ∈ Fin)
29	28	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ¬ (ω ∖ 𝑃) ∈ Fin)
30	5	fin23lem22 9149	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((ω ∖ 𝑃) ⊆ ω ∧ ¬ (ω ∖ 𝑃) ∈ Fin) → 𝑅:ω–1-1-onto→(ω ∖ 𝑃))
31	18, 29, 30	sylancr 695	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → 𝑅:ω–1-1-onto→(ω ∖ 𝑃))
32		f1of 6137	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑅:ω–1-1-onto→(ω ∖ 𝑃) → 𝑅:ω⟶(ω ∖ 𝑃))
33	31, 32	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → 𝑅:ω⟶(ω ∖ 𝑃))
34		simpr 477	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → 𝑏 ∈ dom 𝑅)
35		fdm 6051	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑅:ω⟶(ω ∖ 𝑃) → dom 𝑅 = ω)
36	33, 35	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → dom 𝑅 = ω)
37	34, 36	eleqtrd 2703	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → 𝑏 ∈ ω)
38	33, 37	ffvelrnd 6360	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → (𝑅‘𝑏) ∈ (ω ∖ 𝑃))
39	38	eldifbd 3587	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ¬ (𝑅‘𝑏) ∈ 𝑃)
40	20	eleq2i 2693	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑅‘𝑏) ∈ 𝑃 ↔ (𝑅‘𝑏) ∈ {𝑣 ∈ ω ∣ ∩ ran 𝑈 ⊆ (𝑡‘𝑣)})
41	39, 40	sylnib 318	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ¬ (𝑅‘𝑏) ∈ {𝑣 ∈ ω ∣ ∩ ran 𝑈 ⊆ (𝑡‘𝑣)})
42	38	eldifad 3586	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → (𝑅‘𝑏) ∈ ω)
43		fveq2 6191	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 = (𝑅‘𝑏) → (𝑡‘𝑣) = (𝑡‘(𝑅‘𝑏)))
44	43	sseq2d 3633	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 = (𝑅‘𝑏) → (∩ ran 𝑈 ⊆ (𝑡‘𝑣) ↔ ∩ ran 𝑈 ⊆ (𝑡‘(𝑅‘𝑏))))
45	44	elrab3 3364	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑅‘𝑏) ∈ ω → ((𝑅‘𝑏) ∈ {𝑣 ∈ ω ∣ ∩ ran 𝑈 ⊆ (𝑡‘𝑣)} ↔ ∩ ran 𝑈 ⊆ (𝑡‘(𝑅‘𝑏))))
46	42, 45	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ((𝑅‘𝑏) ∈ {𝑣 ∈ ω ∣ ∩ ran 𝑈 ⊆ (𝑡‘𝑣)} ↔ ∩ ran 𝑈 ⊆ (𝑡‘(𝑅‘𝑏))))
47	41, 46	mtbid 314	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ¬ ∩ ran 𝑈 ⊆ (𝑡‘(𝑅‘𝑏)))
48		fin23lem.a	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑈 = seq𝜔((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢))), ∪ ran 𝑡)
49	48	fin23lem20 9159	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑅‘𝑏) ∈ ω → (∩ ran 𝑈 ⊆ (𝑡‘(𝑅‘𝑏)) ∨ (∩ ran 𝑈 ∩ (𝑡‘(𝑅‘𝑏))) = ∅))
50	42, 49	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → (∩ ran 𝑈 ⊆ (𝑡‘(𝑅‘𝑏)) ∨ (∩ ran 𝑈 ∩ (𝑡‘(𝑅‘𝑏))) = ∅))
51		orel1 397	. . . . . . . . . . . . . . . . 17 ⊢ (¬ ∩ ran 𝑈 ⊆ (𝑡‘(𝑅‘𝑏)) → ((∩ ran 𝑈 ⊆ (𝑡‘(𝑅‘𝑏)) ∨ (∩ ran 𝑈 ∩ (𝑡‘(𝑅‘𝑏))) = ∅) → (∩ ran 𝑈 ∩ (𝑡‘(𝑅‘𝑏))) = ∅))
52	47, 50, 51	sylc 65	. . . . . . . . . . . . . . . 16 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → (∩ ran 𝑈 ∩ (𝑡‘(𝑅‘𝑏))) = ∅)
53	17, 52	syl5eq 2668	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ((𝑡‘(𝑅‘𝑏)) ∩ ∩ ran 𝑈) = ∅)
54		disj 4017	. . . . . . . . . . . . . . 15 ⊢ (((𝑡‘(𝑅‘𝑏)) ∩ ∩ ran 𝑈) = ∅ ↔ ∀𝑎 ∈ (𝑡‘(𝑅‘𝑏)) ¬ 𝑎 ∈ ∩ ran 𝑈)
55	53, 54	sylib 208	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ∀𝑎 ∈ (𝑡‘(𝑅‘𝑏)) ¬ 𝑎 ∈ ∩ ran 𝑈)
56		rsp 2929	. . . . . . . . . . . . . 14 ⊢ (∀𝑎 ∈ (𝑡‘(𝑅‘𝑏)) ¬ 𝑎 ∈ ∩ ran 𝑈 → (𝑎 ∈ (𝑡‘(𝑅‘𝑏)) → ¬ 𝑎 ∈ ∩ ran 𝑈))
57	55, 56	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → (𝑎 ∈ (𝑡‘(𝑅‘𝑏)) → ¬ 𝑎 ∈ ∩ ran 𝑈))
58	16, 57	sylbid 230	. . . . . . . . . . . 12 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → (𝑎 ∈ ((𝑡 ∘ 𝑅)‘𝑏) → ¬ 𝑎 ∈ ∩ ran 𝑈))
59	58	ex 450	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ Fin ∧ Fun 𝑡) → (𝑏 ∈ dom 𝑅 → (𝑎 ∈ ((𝑡 ∘ 𝑅)‘𝑏) → ¬ 𝑎 ∈ ∩ ran 𝑈)))
60	12, 59	syl5 34	. . . . . . . . . 10 ⊢ ((𝑃 ∈ Fin ∧ Fun 𝑡) → (𝑏 ∈ dom (𝑡 ∘ 𝑅) → (𝑎 ∈ ((𝑡 ∘ 𝑅)‘𝑏) → ¬ 𝑎 ∈ ∩ ran 𝑈)))
61	60	rexlimdv 3030	. . . . . . . . 9 ⊢ ((𝑃 ∈ Fin ∧ Fun 𝑡) → (∃𝑏 ∈ dom (𝑡 ∘ 𝑅)𝑎 ∈ ((𝑡 ∘ 𝑅)‘𝑏) → ¬ 𝑎 ∈ ∩ ran 𝑈))
62	10, 61	sylbid 230	. . . . . . . 8 ⊢ ((𝑃 ∈ Fin ∧ Fun 𝑡) → (𝑎 ∈ ∪ ran (𝑡 ∘ 𝑅) → ¬ 𝑎 ∈ ∩ ran 𝑈))
63	62	ralrimiv 2965	. . . . . . 7 ⊢ ((𝑃 ∈ Fin ∧ Fun 𝑡) → ∀𝑎 ∈ ∪ ran (𝑡 ∘ 𝑅) ¬ 𝑎 ∈ ∩ ran 𝑈)
64		disj 4017	. . . . . . 7 ⊢ ((∪ ran (𝑡 ∘ 𝑅) ∩ ∩ ran 𝑈) = ∅ ↔ ∀𝑎 ∈ ∪ ran (𝑡 ∘ 𝑅) ¬ 𝑎 ∈ ∩ ran 𝑈)
65	63, 64	sylibr 224	. . . . . 6 ⊢ ((𝑃 ∈ Fin ∧ Fun 𝑡) → (∪ ran (𝑡 ∘ 𝑅) ∩ ∩ ran 𝑈) = ∅)
66		rneq 5351	. . . . . . . . 9 ⊢ (𝑍 = (𝑡 ∘ 𝑅) → ran 𝑍 = ran (𝑡 ∘ 𝑅))
67	66	unieqd 4446	. . . . . . . 8 ⊢ (𝑍 = (𝑡 ∘ 𝑅) → ∪ ran 𝑍 = ∪ ran (𝑡 ∘ 𝑅))
68	67	ineq1d 3813	. . . . . . 7 ⊢ (𝑍 = (𝑡 ∘ 𝑅) → (∪ ran 𝑍 ∩ ∩ ran 𝑈) = (∪ ran (𝑡 ∘ 𝑅) ∩ ∩ ran 𝑈))
69	68	eqeq1d 2624	. . . . . 6 ⊢ (𝑍 = (𝑡 ∘ 𝑅) → ((∪ ran 𝑍 ∩ ∩ ran 𝑈) = ∅ ↔ (∪ ran (𝑡 ∘ 𝑅) ∩ ∩ ran 𝑈) = ∅))
70	65, 69	syl5ibr 236	. . . . 5 ⊢ (𝑍 = (𝑡 ∘ 𝑅) → ((𝑃 ∈ Fin ∧ Fun 𝑡) → (∪ ran 𝑍 ∩ ∩ ran 𝑈) = ∅))
71	70	expd 452	. . . 4 ⊢ (𝑍 = (𝑡 ∘ 𝑅) → (𝑃 ∈ Fin → (Fun 𝑡 → (∪ ran 𝑍 ∩ ∩ ran 𝑈) = ∅)))
72	71	impcom 446	. . 3 ⊢ ((𝑃 ∈ Fin ∧ 𝑍 = (𝑡 ∘ 𝑅)) → (Fun 𝑡 → (∪ ran 𝑍 ∩ ∩ ran 𝑈) = ∅))
73		rneq 5351	. . . . . . . 8 ⊢ (𝑍 = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄) → ran 𝑍 = ran ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄))
74	73	unieqd 4446	. . . . . . 7 ⊢ (𝑍 = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄) → ∪ ran 𝑍 = ∪ ran ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄))
75	74	ineq1d 3813	. . . . . 6 ⊢ (𝑍 = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄) → (∪ ran 𝑍 ∩ ∩ ran 𝑈) = (∪ ran ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄) ∩ ∩ ran 𝑈))
76		rncoss 5386	. . . . . . . 8 ⊢ ran ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄) ⊆ ran (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈))
77	76	unissi 4461	. . . . . . 7 ⊢ ∪ ran ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄) ⊆ ∪ ran (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈))
78		disj 4017	. . . . . . . 8 ⊢ ((∪ ran (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∩ ∩ ran 𝑈) = ∅ ↔ ∀𝑎 ∈ ∪ ran (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ¬ 𝑎 ∈ ∩ ran 𝑈)
79		eluniab 4447	. . . . . . . . . 10 ⊢ (𝑎 ∈ ∪ {𝑏 ∣ ∃𝑧 ∈ 𝑃 𝑏 = ((𝑡‘𝑧) ∖ ∩ ran 𝑈)} ↔ ∃𝑏(𝑎 ∈ 𝑏 ∧ ∃𝑧 ∈ 𝑃 𝑏 = ((𝑡‘𝑧) ∖ ∩ ran 𝑈)))
80		eleq2 2690	. . . . . . . . . . . . . 14 ⊢ (𝑏 = ((𝑡‘𝑧) ∖ ∩ ran 𝑈) → (𝑎 ∈ 𝑏 ↔ 𝑎 ∈ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)))
81		eldifn 3733	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ ((𝑡‘𝑧) ∖ ∩ ran 𝑈) → ¬ 𝑎 ∈ ∩ ran 𝑈)
82	80, 81	syl6bi 243	. . . . . . . . . . . . 13 ⊢ (𝑏 = ((𝑡‘𝑧) ∖ ∩ ran 𝑈) → (𝑎 ∈ 𝑏 → ¬ 𝑎 ∈ ∩ ran 𝑈))
83	82	rexlimivw 3029	. . . . . . . . . . . 12 ⊢ (∃𝑧 ∈ 𝑃 𝑏 = ((𝑡‘𝑧) ∖ ∩ ran 𝑈) → (𝑎 ∈ 𝑏 → ¬ 𝑎 ∈ ∩ ran 𝑈))
84	83	impcom 446	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ 𝑏 ∧ ∃𝑧 ∈ 𝑃 𝑏 = ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) → ¬ 𝑎 ∈ ∩ ran 𝑈)
85	84	exlimiv 1858	. . . . . . . . . 10 ⊢ (∃𝑏(𝑎 ∈ 𝑏 ∧ ∃𝑧 ∈ 𝑃 𝑏 = ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) → ¬ 𝑎 ∈ ∩ ran 𝑈)
86	79, 85	sylbi 207	. . . . . . . . 9 ⊢ (𝑎 ∈ ∪ {𝑏 ∣ ∃𝑧 ∈ 𝑃 𝑏 = ((𝑡‘𝑧) ∖ ∩ ran 𝑈)} → ¬ 𝑎 ∈ ∩ ran 𝑈)
87		eqid 2622	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) = (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈))
88	87	rnmpt 5371	. . . . . . . . . 10 ⊢ ran (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) = {𝑏 ∣ ∃𝑧 ∈ 𝑃 𝑏 = ((𝑡‘𝑧) ∖ ∩ ran 𝑈)}
89	88	unieqi 4445	. . . . . . . . 9 ⊢ ∪ ran (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) = ∪ {𝑏 ∣ ∃𝑧 ∈ 𝑃 𝑏 = ((𝑡‘𝑧) ∖ ∩ ran 𝑈)}
90	86, 89	eleq2s 2719	. . . . . . . 8 ⊢ (𝑎 ∈ ∪ ran (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) → ¬ 𝑎 ∈ ∩ ran 𝑈)
91	78, 90	mprgbir 2927	. . . . . . 7 ⊢ (∪ ran (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∩ ∩ ran 𝑈) = ∅
92		ssdisj 4026	. . . . . . 7 ⊢ ((∪ ran ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄) ⊆ ∪ ran (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∧ (∪ ran (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∩ ∩ ran 𝑈) = ∅) → (∪ ran ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄) ∩ ∩ ran 𝑈) = ∅)
93	77, 91, 92	mp2an 708	. . . . . 6 ⊢ (∪ ran ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄) ∩ ∩ ran 𝑈) = ∅
94	75, 93	syl6eq 2672	. . . . 5 ⊢ (𝑍 = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄) → (∪ ran 𝑍 ∩ ∩ ran 𝑈) = ∅)
95	94	a1d 25	. . . 4 ⊢ (𝑍 = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄) → (Fun 𝑡 → (∪ ran 𝑍 ∩ ∩ ran 𝑈) = ∅))
96	95	adantl 482	. . 3 ⊢ ((¬ 𝑃 ∈ Fin ∧ 𝑍 = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄)) → (Fun 𝑡 → (∪ ran 𝑍 ∩ ∩ ran 𝑈) = ∅))
97	72, 96	jaoi 394	. 2 ⊢ (((𝑃 ∈ Fin ∧ 𝑍 = (𝑡 ∘ 𝑅)) ∨ (¬ 𝑃 ∈ Fin ∧ 𝑍 = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄))) → (Fun 𝑡 → (∪ ran 𝑍 ∩ ∩ ran 𝑈) = ∅))
98	1, 3, 97	mp2b 10	1 ⊢ (Fun 𝑡 → (∪ ran 𝑍 ∩ ∩ ran 𝑈) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  {cab 2608  ∀wral 2912  ∃wrex 2913  {crab 2916  Vcvv 3200   ∖ cdif 3571   ∪ cun 3572   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  ifcif 4086  𝒫 cpw 4158  ∪ cuni 4436  ∩ cint 4475   class class class wbr 4653   ↦ cmpt 4729  dom cdm 5114  ran crn 5115   ∘ ccom 5118  suc csuc 5725  Fun wfun 5882  ⟶wf 5884  –1-1-onto→wf1o 5887  ‘cfv 5888  ℩crio 6610  (class class class)co 6650   ↦ cmpt2 6652  ωcom 7065  seq_𝜔cseqom 7542   ↑_𝑚 cmap 7857   ≈ cen 7952  Fincfn 7955

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-om 7066  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-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765

This theorem is referenced by:  fin23lem31  9165