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Theorem txcmplem2 23869

Description: Lemma for txcmp 23870. (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 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑅 ∈ Comp)
6	1	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑆 ∈ Comp)
7		txcmp.w	. . . . . 6 ⊢ (𝜑 → 𝑊 ⊆ (𝑅 ×_t 𝑆))
8	7	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑊 ⊆ (𝑅 ×_t 𝑆))
9		txcmp.u	. . . . . 6 ⊢ (𝜑 → (𝑋 × 𝑌) = ∪ 𝑊)
10	9	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (𝑋 × 𝑌) = ∪ 𝑊)
11		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ 𝑌)
12	2, 3, 5, 6, 8, 10, 11	txcmplem1 23868	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → ∃𝑢 ∈ 𝑆 (𝑥 ∈ 𝑢 ∧ ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ ∪ 𝑣))
13	12	ralrimiva 3156	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑌 ∃𝑢 ∈ 𝑆 (𝑥 ∈ 𝑢 ∧ ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ ∪ 𝑣))
14		unieq 4881	. . . . 5 ⊢ (𝑣 = (𝑓‘𝑢) → ∪ 𝑣 = ∪ (𝑓‘𝑢))
15	14	sseq2d 3966	. . . 4 ⊢ (𝑣 = (𝑓‘𝑢) → ((𝑋 × 𝑢) ⊆ ∪ 𝑣 ↔ (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))
16	3, 15	cmpcovf 23617	. . 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 596	. 2 ⊢ (𝜑 → ∃𝑤 ∈ (𝒫 𝑆 ∩ Fin)(𝑌 = ∪ 𝑤 ∧ ∃𝑓(𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢))))
18		simprrl 793	. . . . . . . . . . 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 6706	. . . . . . . . . . 11 ⊢ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) → 𝑓 Fn 𝑤)
20		fniunfv 7247	. . . . . . . . . . 11 ⊢ (𝑓 Fn 𝑤 → ∪ 𝑧 ∈ 𝑤 (𝑓‘𝑧) = ∪ ran 𝑓)
21	18, 19, 20	3syl 19	. . . . . . . . . 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 6715	. . . . . . . . . . . 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 4185	. . . . . . . . . . . 12 ⊢ (𝒫 𝑊 ∩ Fin) ⊆ 𝒫 𝑊
24	22, 23	sstrdi 3946	. . . . . . . . . . 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 5064	. . . . . . . . . . 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 3968	. . . . . . . . 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 3457	. . . . . . . . . . 11 ⊢ 𝑤 ∈ V
29		fvex 6895	. . . . . . . . . . 11 ⊢ (𝑓‘𝑧) ∈ V
30	28, 29	iunex 7968	. . . . . . . . . 10 ⊢ ∪ 𝑧 ∈ 𝑤 (𝑓‘𝑧) ∈ V
31	30	elpw 4564	. . . . . . . . 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 4186	. . . . . . . . . 10 ⊢ (𝒫 𝑆 ∩ Fin) ⊆ Fin
34		simplr 781	. . . . . . . . . 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 3932	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → 𝑤 ∈ Fin)
36		inss2 4186	. . . . . . . . . . 11 ⊢ (𝒫 𝑊 ∩ Fin) ⊆ Fin
37		fss 6723	. . . . . . . . . . 11 ⊢ ((𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ (𝒫 𝑊 ∩ Fin) ⊆ Fin) → 𝑓:𝑤⟶Fin)
38	18, 36, 37	sylancl 598	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → 𝑓:𝑤⟶Fin)
39		ffvelcdm 7077	. . . . . . . . . . 11 ⊢ ((𝑓:𝑤⟶Fin ∧ 𝑧 ∈ 𝑤) → (𝑓‘𝑧) ∈ Fin)
40	39	ralrimiva 3156	. . . . . . . . . 10 ⊢ (𝑓:𝑤⟶Fin → ∀𝑧 ∈ 𝑤 (𝑓‘𝑧) ∈ Fin)
41	38, 40	syl 18	. . . . . . . . 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 9313	. . . . . . . . 9 ⊢ ((𝑤 ∈ Fin ∧ ∀𝑧 ∈ 𝑤 (𝑓‘𝑧) ∈ Fin) → ∪ 𝑧 ∈ 𝑤 (𝑓‘𝑧) ∈ Fin)
43	35, 41, 42	syl2anc 596	. . . . . . . 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 4149	. . . . . . 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 783	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → 𝑌 = ∪ 𝑤)
46		uniiun 5021	. . . . . . . . . . . . 13 ⊢ ∪ 𝑤 = ∪ 𝑧 ∈ 𝑤 𝑧
47	45, 46	eqtrdi 2813	. . . . . . . . . . . 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 5689	. . . . . . . . . . 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 5730	. . . . . . . . . . 11 ⊢ (𝑋 × ∪ 𝑧 ∈ 𝑤 𝑧) = ∪ 𝑧 ∈ 𝑤 (𝑋 × 𝑧)
50	48, 49	eqtrdi 2813	. . . . . . . . . 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 794	. . . . . . . . . . . 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 5680	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝑧 → (𝑋 × 𝑢) = (𝑋 × 𝑧))
53		fveq2 6882	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑧 → (𝑓‘𝑢) = (𝑓‘𝑧))
54	53	unieqd 4883	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝑧 → ∪ (𝑓‘𝑢) = ∪ (𝑓‘𝑧))
55	52, 54	sseq12d 3967	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑧 → ((𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢) ↔ (𝑋 × 𝑧) ⊆ ∪ (𝑓‘𝑧)))
56	55	cbvralvw 3242	. . . . . . . . . . . 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 4973	. . . . . . . . . . 11 ⊢ (∀𝑧 ∈ 𝑤 (𝑋 × 𝑧) ⊆ ∪ (𝑓‘𝑧) → ∪ 𝑧 ∈ 𝑤 (𝑋 × 𝑧) ⊆ ∪ 𝑧 ∈ 𝑤 ∪ (𝑓‘𝑧))
59	57, 58	syl 18	. . . . . . . . . 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 3968	. . . . . . . . 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 7080	. . . . . . . . . . . . . 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 3932	. . . . . . . . . . . . 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 4567	. . . . . . . . . . . . 13 ⊢ ((𝑓‘𝑧) ∈ 𝒫 𝑊 → (𝑓‘𝑧) ⊆ 𝑊)
64		uniss 4878	. . . . . . . . . . . . 13 ⊢ ((𝑓‘𝑧) ⊆ 𝑊 → ∪ (𝑓‘𝑧) ⊆ ∪ 𝑊)
65	62, 63, 64	3syl 19	. . . . . . . . . . . 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 743	. . . . . . . . . . . 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 3971	. . . . . . . . . . 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 3156	. . . . . . . . . 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 5007	. . . . . . . . . 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 3951	. . . . . . . 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 4963	. . . . . . . 8 ⊢ ∪ 𝑧 ∈ 𝑤 ∪ (𝑓‘𝑧) = ∪ ∪ 𝑧 ∈ 𝑤 (𝑓‘𝑧)
73	71, 72	eqtrdi 2813	. . . . . . 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 4881	. . . . . . . 8 ⊢ (𝑣 = ∪ 𝑧 ∈ 𝑤 (𝑓‘𝑧) → ∪ 𝑣 = ∪ ∪ 𝑧 ∈ 𝑤 (𝑓‘𝑧))
75	74	rspceeqv 3602	. . . . . . 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 596	. . . . . 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 462	. . . . 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 1966	. . . 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 459	. . 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 3165	. 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 16	1 ⊢ (𝜑 → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑌) = ∪ 𝑣)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∃wex 1812 ∈ wcel 2145 ∀wral 3078 ∃wrex 3088 ∩ cin 3901 ⊆ wss 3902 𝒫 cpw 4560 ∪ cuni 4870 ∪ ciun 4954 × cxp 5657 ran crn 5660 Fn wfn 6532 ⟶wf 6533 ‘cfv 6537 (class class class)co 7416 Fincfn 8955 Compccmp 23612 ×_t ctx 23787

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-ral 3079 df-rex 3089 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7866 df-1st 7989 df-2nd 7990 df-1o 8458 df-2o 8459 df-en 8956 df-dom 8957 df-fin 8959 df-topgen 17532 df-top 23120 df-bases 23172 df-cmp 23613 df-tx 23789

This theorem is used by: txcmp 23870

W3C validator