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Theorem txcmplem2 21445

Description: Lemma for txcmp 21446. (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 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑅 ∈ Comp)
6	1	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑆 ∈ Comp)
7		txcmp.w	. . . . . 6 ⊢ (𝜑 → 𝑊 ⊆ (𝑅 ×_t 𝑆))
8	7	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑊 ⊆ (𝑅 ×_t 𝑆))
9		txcmp.u	. . . . . 6 ⊢ (𝜑 → (𝑋 × 𝑌) = ∪ 𝑊)
10	9	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (𝑋 × 𝑌) = ∪ 𝑊)
11		simpr 477	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ 𝑌)
12	2, 3, 5, 6, 8, 10, 11	txcmplem1 21444	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → ∃𝑢 ∈ 𝑆 (𝑥 ∈ 𝑢 ∧ ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ ∪ 𝑣))
13	12	ralrimiva 2966	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑌 ∃𝑢 ∈ 𝑆 (𝑥 ∈ 𝑢 ∧ ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ ∪ 𝑣))
14		unieq 4444	. . . . 5 ⊢ (𝑣 = (𝑓‘𝑢) → ∪ 𝑣 = ∪ (𝑓‘𝑢))
15	14	sseq2d 3633	. . . 4 ⊢ (𝑣 = (𝑓‘𝑢) → ((𝑋 × 𝑢) ⊆ ∪ 𝑣 ↔ (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))
16	3, 15	cmpcovf 21194	. . 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 693	. 2 ⊢ (𝜑 → ∃𝑤 ∈ (𝒫 𝑆 ∩ Fin)(𝑌 = ∪ 𝑤 ∧ ∃𝑓(𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢))))
18		simprrl 804	. . . . . . . . . . 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 6045	. . . . . . . . . . 11 ⊢ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) → 𝑓 Fn 𝑤)
20		fniunfv 6505	. . . . . . . . . . 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		frn 6053	. . . . . . . . . . . . 13 ⊢ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) → ran 𝑓 ⊆ (𝒫 𝑊 ∩ Fin))
23	18, 22	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → ran 𝑓 ⊆ (𝒫 𝑊 ∩ Fin))
24		inss1 3833	. . . . . . . . . . . 12 ⊢ (𝒫 𝑊 ∩ Fin) ⊆ 𝒫 𝑊
25	23, 24	syl6ss 3615	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → ran 𝑓 ⊆ 𝒫 𝑊)
26		sspwuni 4611	. . . . . . . . . . 11 ⊢ (ran 𝑓 ⊆ 𝒫 𝑊 ↔ ∪ ran 𝑓 ⊆ 𝑊)
27	25, 26	sylib 208	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → ∪ ran 𝑓 ⊆ 𝑊)
28	21, 27	eqsstrd 3639	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → ∪ 𝑧 ∈ 𝑤 (𝑓‘𝑧) ⊆ 𝑊)
29		vex 3203	. . . . . . . . . . 11 ⊢ 𝑤 ∈ V
30		fvex 6201	. . . . . . . . . . 11 ⊢ (𝑓‘𝑧) ∈ V
31	29, 30	iunex 7147	. . . . . . . . . 10 ⊢ ∪ 𝑧 ∈ 𝑤 (𝑓‘𝑧) ∈ V
32	31	elpw 4164	. . . . . . . . 9 ⊢ (∪ 𝑧 ∈ 𝑤 (𝑓‘𝑧) ∈ 𝒫 𝑊 ↔ ∪ 𝑧 ∈ 𝑤 (𝑓‘𝑧) ⊆ 𝑊)
33	28, 32	sylibr 224	. . . . . . . 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)
34		inss2 3834	. . . . . . . . . 10 ⊢ (𝒫 𝑆 ∩ Fin) ⊆ Fin
35		simplr 792	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → 𝑤 ∈ (𝒫 𝑆 ∩ Fin))
36	34, 35	sseldi 3601	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → 𝑤 ∈ Fin)
37		inss2 3834	. . . . . . . . . . 11 ⊢ (𝒫 𝑊 ∩ Fin) ⊆ Fin
38		fss 6056	. . . . . . . . . . 11 ⊢ ((𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ (𝒫 𝑊 ∩ Fin) ⊆ Fin) → 𝑓:𝑤⟶Fin)
39	18, 37, 38	sylancl 694	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → 𝑓:𝑤⟶Fin)
40		ffvelrn 6357	. . . . . . . . . . 11 ⊢ ((𝑓:𝑤⟶Fin ∧ 𝑧 ∈ 𝑤) → (𝑓‘𝑧) ∈ Fin)
41	40	ralrimiva 2966	. . . . . . . . . 10 ⊢ (𝑓:𝑤⟶Fin → ∀𝑧 ∈ 𝑤 (𝑓‘𝑧) ∈ Fin)
42	39, 41	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)
43		iunfi 8254	. . . . . . . . 9 ⊢ ((𝑤 ∈ Fin ∧ ∀𝑧 ∈ 𝑤 (𝑓‘𝑧) ∈ Fin) → ∪ 𝑧 ∈ 𝑤 (𝑓‘𝑧) ∈ Fin)
44	36, 42, 43	syl2anc 693	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → ∪ 𝑧 ∈ 𝑤 (𝑓‘𝑧) ∈ Fin)
45	33, 44	elind 3798	. . . . . . 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))
46		simprl 794	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → 𝑌 = ∪ 𝑤)
47		uniiun 4573	. . . . . . . . . . . . 13 ⊢ ∪ 𝑤 = ∪ 𝑧 ∈ 𝑤 𝑧
48	46, 47	syl6eq 2672	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) → 𝑌 = ∪ 𝑧 ∈ 𝑤 𝑧)
49	48	xpeq2d 5139	. . . . . . . . . . 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))
50		xpiundi 5173	. . . . . . . . . . 11 ⊢ (𝑋 × ∪ 𝑧 ∈ 𝑤 𝑧) = ∪ 𝑧 ∈ 𝑤 (𝑋 × 𝑧)
51	49, 50	syl6eq 2672	. . . . . . . . . 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))
52		simprrr 805	. . . . . . . . . . . 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))
53		xpeq2 5129	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝑧 → (𝑋 × 𝑢) = (𝑋 × 𝑧))
54		fveq2 6191	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑧 → (𝑓‘𝑢) = (𝑓‘𝑧))
55	54	unieqd 4446	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝑧 → ∪ (𝑓‘𝑢) = ∪ (𝑓‘𝑧))
56	53, 55	sseq12d 3634	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑧 → ((𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢) ↔ (𝑋 × 𝑧) ⊆ ∪ (𝑓‘𝑧)))
57	56	cbvralv 3171	. . . . . . . . . . . 12 ⊢ (∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢) ↔ ∀𝑧 ∈ 𝑤 (𝑋 × 𝑧) ⊆ ∪ (𝑓‘𝑧))
58	52, 57	sylib 208	. . . . . . . . . . 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))
59		ss2iun 4536	. . . . . . . . . . 11 ⊢ (∀𝑧 ∈ 𝑤 (𝑋 × 𝑧) ⊆ ∪ (𝑓‘𝑧) → ∪ 𝑧 ∈ 𝑤 (𝑋 × 𝑧) ⊆ ∪ 𝑧 ∈ 𝑤 ∪ (𝑓‘𝑧))
60	58, 59	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))
61	51, 60	eqsstrd 3639	. . . . . . . . 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))
62	18	ffvelrnda 6359	. . . . . . . . . . . . . 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))
63	24, 62	sseldi 3601	. . . . . . . . . . . . 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)
64		elpwi 4168	. . . . . . . . . . . . 13 ⊢ ((𝑓‘𝑧) ∈ 𝒫 𝑊 → (𝑓‘𝑧) ⊆ 𝑊)
65		uniss 4458	. . . . . . . . . . . . 13 ⊢ ((𝑓‘𝑧) ⊆ 𝑊 → ∪ (𝑓‘𝑧) ⊆ ∪ 𝑊)
66	63, 64, 65	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)
67	9	ad3antrrr 766	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = ∪ 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢 ∈ 𝑤 (𝑋 × 𝑢) ⊆ ∪ (𝑓‘𝑢)))) ∧ 𝑧 ∈ 𝑤) → (𝑋 × 𝑌) = ∪ 𝑊)
68	66, 67	sseqtr4d 3642	. . . . . . . . . . 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))
69	68	ralrimiva 2966	. . . . . . . . . 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))
70		iunss 4561	. . . . . . . . . 10 ⊢ (∪ 𝑧 ∈ 𝑤 ∪ (𝑓‘𝑧) ⊆ (𝑋 × 𝑌) ↔ ∀𝑧 ∈ 𝑤 ∪ (𝑓‘𝑧) ⊆ (𝑋 × 𝑌))
71	69, 70	sylibr 224	. . . . . . . . 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))
72	61, 71	eqssd 3620	. . . . . . . 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))
73		iuncom4 4528	. . . . . . . 8 ⊢ ∪ 𝑧 ∈ 𝑤 ∪ (𝑓‘𝑧) = ∪ ∪ 𝑧 ∈ 𝑤 (𝑓‘𝑧)
74	72, 73	syl6eq 2672	. . . . . . 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))
75		unieq 4444	. . . . . . . . 9 ⊢ (𝑣 = ∪ 𝑧 ∈ 𝑤 (𝑓‘𝑧) → ∪ 𝑣 = ∪ ∪ 𝑧 ∈ 𝑤 (𝑓‘𝑧))
76	75	eqeq2d 2632	. . . . . . . 8 ⊢ (𝑣 = ∪ 𝑧 ∈ 𝑤 (𝑓‘𝑧) → ((𝑋 × 𝑌) = ∪ 𝑣 ↔ (𝑋 × 𝑌) = ∪ ∪ 𝑧 ∈ 𝑤 (𝑓‘𝑧)))
77	76	rspcev 3309	. . . . . . 7 ⊢ ((∪ 𝑧 ∈ 𝑤 (𝑓‘𝑧) ∈ (𝒫 𝑊 ∩ Fin) ∧ (𝑋 × 𝑌) = ∪ ∪ 𝑧 ∈ 𝑤 (𝑓‘𝑧)) → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑌) = ∪ 𝑣)
78	45, 74, 77	syl2anc 693	. . . . . 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)
79	78	expr 643	. . . . 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))
80	79	exlimdv 1861	. . . 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))
81	80	expimpd 629	. . 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))
82	81	rexlimdva 3031	. 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))
83	17, 82	mpd 15	1 ⊢ (𝜑 → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑌) = ∪ 𝑣)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∃wex 1704 ∈ wcel 1990 ∀wral 2912 ∃wrex 2913 ∩ cin 3573 ⊆ wss 3574 𝒫 cpw 4158 ∪ cuni 4436 ∪ ciun 4520 × cxp 5112 ran crn 5115 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 Fincfn 7955 Compccmp 21189 ×_t ctx 21363

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-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-iin 4523 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-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-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-fin 7959 df-topgen 16104 df-top 20699 df-bases 20750 df-cmp 21190 df-tx 21365

This theorem is referenced by: txcmp 21446

W3C validator