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Theorem txcmplem2 22247

Description: Lemma for txcmp 22248. (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 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑅 ∈ Comp)
6	1	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑆 ∈ Comp)
7		txcmp.w	. . . . . 6 ⊢ (𝜑 → 𝑊 ⊆ (𝑅 ×_t 𝑆))
8	7	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑊 ⊆ (𝑅 ×_t 𝑆))
9		txcmp.u	. . . . . 6 ⊢ (𝜑 → (𝑋 × 𝑌) = ∪ 𝑊)
10	9	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (𝑋 × 𝑌) = ∪ 𝑊)
11		simpr 488	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ 𝑌)
12	2, 3, 5, 6, 8, 10, 11	txcmplem1 22246	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → ∃𝑢 ∈ 𝑆 (𝑥 ∈ 𝑢 ∧ ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ ∪ 𝑣))
13	12	ralrimiva 3149	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑌 ∃𝑢 ∈ 𝑆 (𝑥 ∈ 𝑢 ∧ ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ ∪ 𝑣))
14		unieq 4811	. . . . 5 ⊢ (𝑣 = (𝑓‘𝑢) → ∪ 𝑣 = ∪ (𝑓‘𝑢))
15	14	sseq2d 3947	. . . 4 ⊢ (𝑣 = (𝑓‘𝑢) → ((𝑋 × 𝑢) ⊆ ∪ 𝑣 ↔ (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))
16	3, 15	cmpcovf 21996	. . 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 587	. 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 6487	. . . . . . . . . . 11 ⊢ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) → 𝑓 Fn 𝑤)
20		fniunfv 6984	. . . . . . . . . . 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 6494	. . . . . . . . . . . 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 4155	. . . . . . . . . . . 12 ⊢ (𝒫 𝑊 ∩ Fin) ⊆ 𝒫 𝑊
24	22, 23	sstrdi 3927	. . . . . . . . . . 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 4985	. . . . . . . . . . 11 ⊢ (ran 𝑓 ⊆ 𝒫 𝑊 ↔ ∪ ran 𝑓 ⊆ 𝑊)
26	24, 25	sylib 221	. . . . . . . . . 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 3953	. . . . . . . . 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 3444	. . . . . . . . . . 11 ⊢ 𝑤 ∈ V
29		fvex 6658	. . . . . . . . . . 11 ⊢ (𝑓‘𝑧) ∈ V
30	28, 29	iunex 7651	. . . . . . . . . 10 ⊢ ∪ 𝑧 ∈ 𝑤 (𝑓‘𝑧) ∈ V
31	30	elpw 4501	. . . . . . . . 9 ⊢ (∪ 𝑧 ∈ 𝑤 (𝑓‘𝑧) ∈ 𝒫 𝑊 ↔ ∪ 𝑧 ∈ 𝑤 (𝑓‘𝑧) ⊆ 𝑊)
32	27, 31	sylibr 237	. . . . . . . 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 4156	. . . . . . . . . 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	sseldi 3913	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → 𝑤 ∈ Fin)
36		inss2 4156	. . . . . . . . . . 11 ⊢ (𝒫 𝑊 ∩ Fin) ⊆ Fin
37		fss 6501	. . . . . . . . . . 11 ⊢ ((𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ (𝒫 𝑊 ∩ Fin) ⊆ Fin) → 𝑓:𝑤⟶Fin)
38	18, 36, 37	sylancl 589	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → 𝑓:𝑤⟶Fin)
39		ffvelrn 6826	. . . . . . . . . . 11 ⊢ ((𝑓:𝑤⟶Fin ∧ 𝑧 ∈ 𝑤) → (𝑓‘𝑧) ∈ Fin)
40	39	ralrimiva 3149	. . . . . . . . . 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 8796	. . . . . . . . 9 ⊢ ((𝑤 ∈ Fin ∧ ∀𝑧 ∈ 𝑤 (𝑓‘𝑧) ∈ Fin) → ∪ 𝑧 ∈ 𝑤 (𝑓‘𝑧) ∈ Fin)
43	35, 41, 42	syl2anc 587	. . . . . . . 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 4121	. . . . . . 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 4945	. . . . . . . . . . . . 13 ⊢ ∪ 𝑤 = ∪ 𝑧 ∈ 𝑤 𝑧
47	45, 46	eqtrdi 2849	. . . . . . . . . . . 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 5549	. . . . . . . . . . 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 5586	. . . . . . . . . . 11 ⊢ (𝑋 × ∪ 𝑧 ∈ 𝑤 𝑧) = ∪ 𝑧 ∈ 𝑤 (𝑋 × 𝑧)
50	48, 49	eqtrdi 2849	. . . . . . . . . 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 5540	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝑧 → (𝑋 × 𝑢) = (𝑋 × 𝑧))
53		fveq2 6645	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑧 → (𝑓‘𝑢) = (𝑓‘𝑧))
54	53	unieqd 4814	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝑧 → ∪ (𝑓‘𝑢) = ∪ (𝑓‘𝑧))
55	52, 54	sseq12d 3948	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑧 → ((𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢) ↔ (𝑋 × 𝑧) ⊆ ∪ (𝑓‘𝑧)))
56	55	cbvralvw 3396	. . . . . . . . . . . 12 ⊢ (∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢) ↔ ∀𝑧 ∈ 𝑤 (𝑋 × 𝑧) ⊆ ∪ (𝑓‘𝑧))
57	51, 56	sylib 221	. . . . . . . . . . 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 4899	. . . . . . . . . . 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 3953	. . . . . . . . 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	ffvelrnda 6828	. . . . . . . . . . . . . 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	sseldi 3913	. . . . . . . . . . . . 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 4506	. . . . . . . . . . . . 13 ⊢ ((𝑓‘𝑧) ∈ 𝒫 𝑊 → (𝑓‘𝑧) ⊆ 𝑊)
64		uniss 4808	. . . . . . . . . . . . 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 3956	. . . . . . . . . . 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 3149	. . . . . . . . . 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 4932	. . . . . . . . . 10 ⊢ (∪ 𝑧 ∈ 𝑤 ∪ (𝑓‘𝑧) ⊆ (𝑋 × 𝑌) ↔ ∀𝑧 ∈ 𝑤 ∪ (𝑓‘𝑧) ⊆ (𝑋 × 𝑌))
70	68, 69	sylibr 237	. . . . . . . . 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 3932	. . . . . . . 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 4889	. . . . . . . 8 ⊢ ∪ 𝑧 ∈ 𝑤 ∪ (𝑓‘𝑧) = ∪ ∪ 𝑧 ∈ 𝑤 (𝑓‘𝑧)
73	71, 72	eqtrdi 2849	. . . . . . 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 4811	. . . . . . . 8 ⊢ (𝑣 = ∪ 𝑧 ∈ 𝑤 (𝑓‘𝑧) → ∪ 𝑣 = ∪ ∪ 𝑧 ∈ 𝑤 (𝑓‘𝑧))
75	74	rspceeqv 3586	. . . . . . 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 587	. . . . . 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 460	. . . . 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 1934	. . . 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 457	. . 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 3243	. 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 399 = wceq 1538 ∃wex 1781 ∈ wcel 2111 ∀wral 3106 ∃wrex 3107 ∩ cin 3880 ⊆ wss 3881 𝒫 cpw 4497 ∪ cuni 4800 ∪ ciun 4881 × cxp 5517 ran crn 5520 Fn wfn 6319 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 Fincfn 8492 Compccmp 21991 ×_t ctx 22165

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-fin 8496 df-topgen 16709 df-top 21499 df-bases 21551 df-cmp 21992 df-tx 22167

This theorem is referenced by: txcmp 22248

W3C validator