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Theorem txcmplem2 23671

Description: Lemma for txcmp 23672. (Contributed by Mario Carneiro, 14-Sep-2014.)

**Hypotheses**
Ref	Expression
txcmp.x	⊢ 𝑋 = ∪ 𝑅
txcmp.y	⊢ 𝑌 = ∪ 𝑆
txcmp.r	⊢ (𝜑 → 𝑅 ∈ Comp)
txcmp.s	⊢ (𝜑 → 𝑆 ∈ Comp)
txcmp.w	⊢ (𝜑 → 𝑊 ⊆ (𝑅 ×_t 𝑆))
txcmp.u	⊢ (𝜑 → (𝑋 × 𝑌) = ∪ 𝑊)

**Assertion**
Ref	Expression
txcmplem2	⊢ (𝜑 → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑌) = ∪ 𝑣)

Distinct variable groups: 𝑣,𝑆 𝑣,𝑌 𝑣,𝑊 𝑣,𝑋 Allowed substitution hints: 𝜑(𝑣) 𝑅(𝑣)

Proof of Theorem txcmplem2 Dummy variables 𝑓 𝑢 𝑥 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		txcmp.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ Comp)
2		txcmp.x	. . . . 5 ⊢ 𝑋 = ∪ 𝑅
3		txcmp.y	. . . . 5 ⊢ 𝑌 = ∪ 𝑆
4		txcmp.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Comp)
5	4	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑅 ∈ Comp)
6	1	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑆 ∈ Comp)
7		txcmp.w	. . . . . 6 ⊢ (𝜑 → 𝑊 ⊆ (𝑅 ×_t 𝑆))
8	7	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑊 ⊆ (𝑅 ×_t 𝑆))
9		txcmp.u	. . . . . 6 ⊢ (𝜑 → (𝑋 × 𝑌) = ∪ 𝑊)
10	9	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (𝑋 × 𝑌) = ∪ 𝑊)
11		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ 𝑌)
12	2, 3, 5, 6, 8, 10, 11	txcmplem1 23670	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → ∃𝑢 ∈ 𝑆 (𝑥 ∈ 𝑢 ∧ ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ ∪ 𝑣))
13	12	ralrimiva 3152	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑌 ∃𝑢 ∈ 𝑆 (𝑥 ∈ 𝑢 ∧ ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ ∪ 𝑣))
14		unieq 4942	. . . . 5 ⊢ (𝑣 = (𝑓‘𝑢) → ∪ 𝑣 = ∪ (𝑓‘𝑢))
15	14	sseq2d 4041	. . . 4 ⊢ (𝑣 = (𝑓‘𝑢) → ((𝑋 × 𝑢) ⊆ ∪ 𝑣 ↔ (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))
16	3, 15	cmpcovf 23420	. . 3 ⊢ ((𝑆 ∈ Comp ∧ ∀𝑥 ∈ 𝑌 ∃𝑢 ∈ 𝑆 (𝑥 ∈ 𝑢 ∧ ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ ∪ 𝑣)) → ∃𝑤 ∈ (𝒫 𝑆 ∩ Fin)(𝑌 = ∪ 𝑤 ∧ ∃𝑓(𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢))))
17	1, 13, 16	syl2anc 583	. 2 ⊢ (𝜑 → ∃𝑤 ∈ (𝒫 𝑆 ∩ Fin)(𝑌 = ∪ 𝑤 ∧ ∃𝑓(𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢))))
18		simprrl 780	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → 𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin))
19		ffn 6747	. . . . . . . . . . 11 ⊢ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) → 𝑓 Fn 𝑤)
20		fniunfv 7284	. . . . . . . . . . 11 ⊢ (𝑓 Fn 𝑤 → ∪ 𝑧 ∈ 𝑤 (𝑓‘𝑧) = ∪ ran 𝑓)
21	18, 19, 20	3syl 18	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → ∪ 𝑧 ∈ 𝑤 (𝑓‘𝑧) = ∪ ran 𝑓)
22	18	frnd 6755	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → ran 𝑓 ⊆ (𝒫 𝑊 ∩ Fin))
23		inss1 4258	. . . . . . . . . . . 12 ⊢ (𝒫 𝑊 ∩ Fin) ⊆ 𝒫 𝑊
24	22, 23	sstrdi 4021	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → ran 𝑓 ⊆ 𝒫 𝑊)
25		sspwuni 5123	. . . . . . . . . . 11 ⊢ (ran 𝑓 ⊆ 𝒫 𝑊 ↔ ∪ ran 𝑓 ⊆ 𝑊)
26	24, 25	sylib 218	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → ∪ ran 𝑓 ⊆ 𝑊)
27	21, 26	eqsstrd 4047	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → ∪ 𝑧 ∈ 𝑤 (𝑓‘𝑧) ⊆ 𝑊)
28		vex 3492	. . . . . . . . . . 11 ⊢ 𝑤 ∈ V
29		fvex 6933	. . . . . . . . . . 11 ⊢ (𝑓‘𝑧) ∈ V
30	28, 29	iunex 8009	. . . . . . . . . 10 ⊢ ∪ 𝑧 ∈ 𝑤 (𝑓‘𝑧) ∈ V
31	30	elpw 4626	. . . . . . . . 9 ⊢ (∪ 𝑧 ∈ 𝑤 (𝑓‘𝑧) ∈ 𝒫 𝑊 ↔ ∪ 𝑧 ∈ 𝑤 (𝑓‘𝑧) ⊆ 𝑊)
32	27, 31	sylibr 234	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → ∪ 𝑧 ∈ 𝑤 (𝑓‘𝑧) ∈ 𝒫 𝑊)
33		inss2 4259	. . . . . . . . . 10 ⊢ (𝒫 𝑆 ∩ Fin) ⊆ Fin
34		simplr 768	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → 𝑤 ∈ (𝒫 𝑆 ∩ Fin))
35	33, 34	sselid 4006	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → 𝑤 ∈ Fin)
36		inss2 4259	. . . . . . . . . . 11 ⊢ (𝒫 𝑊 ∩ Fin) ⊆ Fin
37		fss 6763	. . . . . . . . . . 11 ⊢ ((𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ (𝒫 𝑊 ∩ Fin) ⊆ Fin) → 𝑓:𝑤⟶Fin)
38	18, 36, 37	sylancl 585	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → 𝑓:𝑤⟶Fin)
39		ffvelcdm 7115	. . . . . . . . . . 11 ⊢ ((𝑓:𝑤⟶Fin ∧ 𝑧 ∈ 𝑤) → (𝑓‘𝑧) ∈ Fin)
40	39	ralrimiva 3152	. . . . . . . . . 10 ⊢ (𝑓:𝑤⟶Fin → ∀𝑧 ∈ 𝑤 (𝑓‘𝑧) ∈ Fin)
41	38, 40	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → ∀𝑧 ∈ 𝑤 (𝑓‘𝑧) ∈ Fin)
42		iunfi 9411	. . . . . . . . 9 ⊢ ((𝑤 ∈ Fin ∧ ∀𝑧 ∈ 𝑤 (𝑓‘𝑧) ∈ Fin) → ∪ 𝑧 ∈ 𝑤 (𝑓‘𝑧) ∈ Fin)
43	35, 41, 42	syl2anc 583	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → ∪ 𝑧 ∈ 𝑤 (𝑓‘𝑧) ∈ Fin)
44	32, 43	elind 4223	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → ∪ 𝑧 ∈ 𝑤 (𝑓‘𝑧) ∈ (𝒫 𝑊 ∩ Fin))
45		simprl 770	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → 𝑌 = ∪ 𝑤)
46		uniiun 5081	. . . . . . . . . . . . 13 ⊢ ∪ 𝑤 = ∪ 𝑧 ∈ 𝑤 𝑧
47	45, 46	eqtrdi 2796	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → 𝑌 = ∪ 𝑧 ∈ 𝑤 𝑧)
48	47	xpeq2d 5730	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → (𝑋 × 𝑌) = (𝑋 × ∪ 𝑧 ∈ 𝑤 𝑧))
49		xpiundi 5770	. . . . . . . . . . 11 ⊢ (𝑋 × ∪ 𝑧 ∈ 𝑤 𝑧) = ∪ 𝑧 ∈ 𝑤 (𝑋 × 𝑧)
50	48, 49	eqtrdi 2796	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → (𝑋 × 𝑌) = ∪ 𝑧 ∈ 𝑤 (𝑋 × 𝑧))
51		simprrr 781	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢))
52		xpeq2 5721	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝑧 → (𝑋 × 𝑢) = (𝑋 × 𝑧))
53		fveq2 6920	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑧 → (𝑓‘𝑢) = (𝑓‘𝑧))
54	53	unieqd 4944	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝑧 → ∪ (𝑓‘𝑢) = ∪ (𝑓‘𝑧))
55	52, 54	sseq12d 4042	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑧 → ((𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢) ↔ (𝑋 × 𝑧) ⊆ ∪ (𝑓‘𝑧)))
56	55	cbvralvw 3243	. . . . . . . . . . . 12 ⊢ (∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢) ↔ ∀𝑧 ∈ 𝑤 (𝑋 × 𝑧) ⊆ ∪ (𝑓‘𝑧))
57	51, 56	sylib 218	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → ∀𝑧 ∈ 𝑤 (𝑋 × 𝑧) ⊆ ∪ (𝑓‘𝑧))
58		ss2iun 5033	. . . . . . . . . . 11 ⊢ (∀𝑧 ∈ 𝑤 (𝑋 × 𝑧) ⊆ ∪ (𝑓‘𝑧) → ∪ 𝑧 ∈ 𝑤 (𝑋 × 𝑧) ⊆ ∪ 𝑧 ∈ 𝑤 ∪ (𝑓‘𝑧))
59	57, 58	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → ∪ 𝑧 ∈ 𝑤 (𝑋 × 𝑧) ⊆ ∪ 𝑧 ∈ 𝑤 ∪ (𝑓‘𝑧))
60	50, 59	eqsstrd 4047	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → (𝑋 × 𝑌) ⊆ ∪ 𝑧 ∈ 𝑤 ∪ (𝑓‘𝑧))
61	18	ffvelcdmda 7118	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) ∧ 𝑧 ∈ 𝑤) → (𝑓‘𝑧) ∈ (𝒫 𝑊 ∩ Fin))
62	23, 61	sselid 4006	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) ∧ 𝑧 ∈ 𝑤) → (𝑓‘𝑧) ∈ 𝒫 𝑊)
63		elpwi 4629	. . . . . . . . . . . . 13 ⊢ ((𝑓‘𝑧) ∈ 𝒫 𝑊 → (𝑓‘𝑧) ⊆ 𝑊)
64		uniss 4939	. . . . . . . . . . . . 13 ⊢ ((𝑓‘𝑧) ⊆ 𝑊 → ∪ (𝑓‘𝑧) ⊆ ∪ 𝑊)
65	62, 63, 64	3syl 18	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) ∧ 𝑧 ∈ 𝑤) → ∪ (𝑓‘𝑧) ⊆ ∪ 𝑊)
66	9	ad3antrrr 729	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) ∧ 𝑧 ∈ 𝑤) → (𝑋 × 𝑌) = ∪ 𝑊)
67	65, 66	sseqtrrd 4050	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) ∧ 𝑧 ∈ 𝑤) → ∪ (𝑓‘𝑧) ⊆ (𝑋 × 𝑌))
68	67	ralrimiva 3152	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → ∀𝑧 ∈ 𝑤 ∪ (𝑓‘𝑧) ⊆ (𝑋 × 𝑌))
69		iunss 5068	. . . . . . . . . 10 ⊢ (∪ 𝑧 ∈ 𝑤 ∪ (𝑓‘𝑧) ⊆ (𝑋 × 𝑌) ↔ ∀𝑧 ∈ 𝑤 ∪ (𝑓‘𝑧) ⊆ (𝑋 × 𝑌))
70	68, 69	sylibr 234	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → ∪ 𝑧 ∈ 𝑤 ∪ (𝑓‘𝑧) ⊆ (𝑋 × 𝑌))
71	60, 70	eqssd 4026	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → (𝑋 × 𝑌) = ∪ 𝑧 ∈ 𝑤 ∪ (𝑓‘𝑧))
72		iuncom4 5023	. . . . . . . 8 ⊢ ∪ 𝑧 ∈ 𝑤 ∪ (𝑓‘𝑧) = ∪ ∪ 𝑧 ∈ 𝑤 (𝑓‘𝑧)
73	71, 72	eqtrdi 2796	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → (𝑋 × 𝑌) = ∪ ∪ 𝑧 ∈ 𝑤 (𝑓‘𝑧))
74		unieq 4942	. . . . . . . 8 ⊢ (𝑣 = ∪ 𝑧 ∈ 𝑤 (𝑓‘𝑧) → ∪ 𝑣 = ∪ ∪ 𝑧 ∈ 𝑤 (𝑓‘𝑧))
75	74	rspceeqv 3658	. . . . . . 7 ⊢ ((∪ 𝑧 ∈ 𝑤 (𝑓‘𝑧) ∈ (𝒫 𝑊 ∩ Fin) ∧ (𝑋 × 𝑌) = ∪ ∪ 𝑧 ∈ 𝑤 (𝑓‘𝑧)) → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑌) = ∪ 𝑣)
76	44, 73, 75	syl2anc 583	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑌) = ∪ 𝑣)
77	76	expr 456	. . . . 5 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ 𝑌 = ∪ 𝑤) → ((𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)) → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑌) = ∪ 𝑣))
78	77	exlimdv 1932	. . . 4 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ 𝑌 = ∪ 𝑤) → (∃𝑓(𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)) → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑌) = ∪ 𝑣))
79	78	expimpd 453	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) → ((𝑌 = ∪ 𝑤 ∧ ∃𝑓(𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢))) → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑌) = ∪ 𝑣))
80	79	rexlimdva 3161	. 2 ⊢ (𝜑 → (∃𝑤 ∈ (𝒫 𝑆 ∩ Fin)(𝑌 = ∪ 𝑤 ∧ ∃𝑓(𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢))) → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑌) = ∪ 𝑣))
81	17, 80	mpd 15	1 ⊢ (𝜑 → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑌) = ∪ 𝑣)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∃wex 1777 ∈ wcel 2108 ∀wral 3067 ∃wrex 3076 ∩ cin 3975 ⊆ wss 3976 𝒫 cpw 4622 ∪ cuni 4931 ∪ ciun 5015 × cxp 5698 ran crn 5701 Fn wfn 6568 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 Fincfn 9003 Compccmp 23415 ×_t ctx 23589

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-1o 8522 df-2o 8523 df-en 9004 df-dom 9005 df-fin 9007 df-topgen 17503 df-top 22921 df-bases 22974 df-cmp 23416 df-tx 23591

This theorem is referenced by: txcmp 23672

W3C validator