Theorem txcmplem1 22863

Description: Lemma for txcmp 22865. (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)
txcmp.a	⊢ (𝜑 → 𝐴 ∈ 𝑌)

Assertion

Ref	Expression
txcmplem1	⊢ (𝜑 → ∃𝑢 ∈ 𝑆 (𝐴 ∈ 𝑢 ∧ ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ ∪ 𝑣))

Distinct variable groups:   𝑢,𝐴   𝑣,𝑢,𝑆   𝑢,𝑌,𝑣   𝑢,𝑊,𝑣   𝑢,𝑋,𝑣   𝜑,𝑢   𝑢,𝑅

Allowed substitution hints:   𝜑(𝑣)   𝐴(𝑣)   𝑅(𝑣)

Proof of Theorem txcmplem1

Dummy variables 𝑓 𝑘 𝑟 𝑠 𝑡 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		txcmp.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ Comp)
2		id 22	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑋 → 𝑥 ∈ 𝑋)
3		txcmp.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝑌)
4		opelxpi 5642	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → ⟨𝑥, 𝐴⟩ ∈ (𝑋 × 𝑌))
5	2, 3, 4	syl2anr 597	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ⟨𝑥, 𝐴⟩ ∈ (𝑋 × 𝑌))
6		txcmp.u	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 × 𝑌) = ∪ 𝑊)
7	6	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑋 × 𝑌) = ∪ 𝑊)
8	5, 7	eleqtrd 2840	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ⟨𝑥, 𝐴⟩ ∈ ∪ 𝑊)
9		eluni2 4852	. . . . . . 7 ⊢ (⟨𝑥, 𝐴⟩ ∈ ∪ 𝑊 ↔ ∃𝑘 ∈ 𝑊 ⟨𝑥, 𝐴⟩ ∈ 𝑘)
10	8, 9	sylib 217	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∃𝑘 ∈ 𝑊 ⟨𝑥, 𝐴⟩ ∈ 𝑘)
11		txcmp.w	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑊 ⊆ (𝑅 ×t 𝑆))
12	11	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑊 ⊆ (𝑅 ×t 𝑆))
13	12	sselda 3930	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝑊) → 𝑘 ∈ (𝑅 ×t 𝑆))
14		txcmp.s	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑆 ∈ Comp)
15		eltx 22790	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) → (𝑘 ∈ (𝑅 ×t 𝑆) ↔ ∀𝑦 ∈ 𝑘 ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (𝑦 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)))
16	1, 14, 15	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑘 ∈ (𝑅 ×t 𝑆) ↔ ∀𝑦 ∈ 𝑘 ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (𝑦 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)))
17	16	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑘 ∈ (𝑅 ×t 𝑆) ↔ ∀𝑦 ∈ 𝑘 ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (𝑦 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)))
18	17	biimpa 477	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (𝑅 ×t 𝑆)) → ∀𝑦 ∈ 𝑘 ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (𝑦 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘))
19	13, 18	syldan 591	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝑊) → ∀𝑦 ∈ 𝑘 ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (𝑦 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘))
20		eleq1 2825	. . . . . . . . . . . 12 ⊢ (𝑦 = ⟨𝑥, 𝐴⟩ → (𝑦 ∈ (𝑟 × 𝑠) ↔ ⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠)))
21	20	anbi1d 630	. . . . . . . . . . 11 ⊢ (𝑦 = ⟨𝑥, 𝐴⟩ → ((𝑦 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘) ↔ (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)))
22	21	2rexbidv 3210	. . . . . . . . . 10 ⊢ (𝑦 = ⟨𝑥, 𝐴⟩ → (∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (𝑦 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘) ↔ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)))
23	22	rspccv 3567	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝑘 ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (𝑦 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘) → (⟨𝑥, 𝐴⟩ ∈ 𝑘 → ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)))
24	19, 23	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝑊) → (⟨𝑥, 𝐴⟩ ∈ 𝑘 → ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)))
25		opelxp1 5646	. . . . . . . . . . . . 13 ⊢ (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) → 𝑥 ∈ 𝑟)
26	25	ad2antrl 725	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝑊) ∧ (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)) → 𝑥 ∈ 𝑟)
27		opelxp2 5647	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) → 𝐴 ∈ 𝑠)
28	27	ad2antrl 725	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝑊) ∧ (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)) → 𝐴 ∈ 𝑠)
29	28	snssd 4752	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝑊) ∧ (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)) → {𝐴} ⊆ 𝑠)
30		xpss2 5625	. . . . . . . . . . . . . 14 ⊢ ({𝐴} ⊆ 𝑠 → (𝑟 × {𝐴}) ⊆ (𝑟 × 𝑠))
31	29, 30	syl 17	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝑊) ∧ (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)) → (𝑟 × {𝐴}) ⊆ (𝑟 × 𝑠))
32		simprr 770	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝑊) ∧ (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)) → (𝑟 × 𝑠) ⊆ 𝑘)
33	31, 32	sstrd 3940	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝑊) ∧ (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)) → (𝑟 × {𝐴}) ⊆ 𝑘)
34	26, 33	jca 512	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝑊) ∧ (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)) → (𝑥 ∈ 𝑟 ∧ (𝑟 × {𝐴}) ⊆ 𝑘))
35	34	ex 413	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝑊) → ((⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘) → (𝑥 ∈ 𝑟 ∧ (𝑟 × {𝐴}) ⊆ 𝑘)))
36	35	rexlimdvw 3154	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝑊) → (∃𝑠 ∈ 𝑆 (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘) → (𝑥 ∈ 𝑟 ∧ (𝑟 × {𝐴}) ⊆ 𝑘)))
37	36	reximdv 3164	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝑊) → (∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘) → ∃𝑟 ∈ 𝑅 (𝑥 ∈ 𝑟 ∧ (𝑟 × {𝐴}) ⊆ 𝑘)))
38	24, 37	syld 47	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝑊) → (⟨𝑥, 𝐴⟩ ∈ 𝑘 → ∃𝑟 ∈ 𝑅 (𝑥 ∈ 𝑟 ∧ (𝑟 × {𝐴}) ⊆ 𝑘)))
39	38	reximdva 3162	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (∃𝑘 ∈ 𝑊 ⟨𝑥, 𝐴⟩ ∈ 𝑘 → ∃𝑘 ∈ 𝑊 ∃𝑟 ∈ 𝑅 (𝑥 ∈ 𝑟 ∧ (𝑟 × {𝐴}) ⊆ 𝑘)))
40	10, 39	mpd 15	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∃𝑘 ∈ 𝑊 ∃𝑟 ∈ 𝑅 (𝑥 ∈ 𝑟 ∧ (𝑟 × {𝐴}) ⊆ 𝑘))
41		rexcom 3270	. . . . . 6 ⊢ (∃𝑘 ∈ 𝑊 ∃𝑟 ∈ 𝑅 (𝑥 ∈ 𝑟 ∧ (𝑟 × {𝐴}) ⊆ 𝑘) ↔ ∃𝑟 ∈ 𝑅 ∃𝑘 ∈ 𝑊 (𝑥 ∈ 𝑟 ∧ (𝑟 × {𝐴}) ⊆ 𝑘))
42		r19.42v 3184	. . . . . . 7 ⊢ (∃𝑘 ∈ 𝑊 (𝑥 ∈ 𝑟 ∧ (𝑟 × {𝐴}) ⊆ 𝑘) ↔ (𝑥 ∈ 𝑟 ∧ ∃𝑘 ∈ 𝑊 (𝑟 × {𝐴}) ⊆ 𝑘))
43	42	rexbii 3094	. . . . . 6 ⊢ (∃𝑟 ∈ 𝑅 ∃𝑘 ∈ 𝑊 (𝑥 ∈ 𝑟 ∧ (𝑟 × {𝐴}) ⊆ 𝑘) ↔ ∃𝑟 ∈ 𝑅 (𝑥 ∈ 𝑟 ∧ ∃𝑘 ∈ 𝑊 (𝑟 × {𝐴}) ⊆ 𝑘))
44	41, 43	bitri 274	. . . . 5 ⊢ (∃𝑘 ∈ 𝑊 ∃𝑟 ∈ 𝑅 (𝑥 ∈ 𝑟 ∧ (𝑟 × {𝐴}) ⊆ 𝑘) ↔ ∃𝑟 ∈ 𝑅 (𝑥 ∈ 𝑟 ∧ ∃𝑘 ∈ 𝑊 (𝑟 × {𝐴}) ⊆ 𝑘))
45	40, 44	sylib 217	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∃𝑟 ∈ 𝑅 (𝑥 ∈ 𝑟 ∧ ∃𝑘 ∈ 𝑊 (𝑟 × {𝐴}) ⊆ 𝑘))
46	45	ralrimiva 3140	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∃𝑟 ∈ 𝑅 (𝑥 ∈ 𝑟 ∧ ∃𝑘 ∈ 𝑊 (𝑟 × {𝐴}) ⊆ 𝑘))
47		txcmp.x	. . . 4 ⊢ 𝑋 = ∪ 𝑅
48		sseq2 3956	. . . 4 ⊢ (𝑘 = (𝑓‘𝑟) → ((𝑟 × {𝐴}) ⊆ 𝑘 ↔ (𝑟 × {𝐴}) ⊆ (𝑓‘𝑟)))
49	47, 48	cmpcovf 22613	. . 3 ⊢ ((𝑅 ∈ Comp ∧ ∀𝑥 ∈ 𝑋 ∃𝑟 ∈ 𝑅 (𝑥 ∈ 𝑟 ∧ ∃𝑘 ∈ 𝑊 (𝑟 × {𝐴}) ⊆ 𝑘)) → ∃𝑡 ∈ (𝒫 𝑅 ∩ Fin)(𝑋 = ∪ 𝑡 ∧ ∃𝑓(𝑓:𝑡⟶𝑊 ∧ ∀𝑟 ∈ 𝑡 (𝑟 × {𝐴}) ⊆ (𝑓‘𝑟))))
50	1, 46, 49	syl2anc 584	. 2 ⊢ (𝜑 → ∃𝑡 ∈ (𝒫 𝑅 ∩ Fin)(𝑋 = ∪ 𝑡 ∧ ∃𝑓(𝑓:𝑡⟶𝑊 ∧ ∀𝑟 ∈ 𝑡 (𝑟 × {𝐴}) ⊆ (𝑓‘𝑟))))
51		txcmp.y	. . . . . . . 8 ⊢ 𝑌 = ∪ 𝑆
52	1	ad2antrr 723	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑊 ∧ ∀𝑟 ∈ 𝑡 (𝑟 × {𝐴}) ⊆ (𝑓‘𝑟)))) → 𝑅 ∈ Comp)
53		cmptop 22617	. . . . . . . . . 10 ⊢ (𝑆 ∈ Comp → 𝑆 ∈ Top)
54	14, 53	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ Top)
55	54	ad2antrr 723	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑊 ∧ ∀𝑟 ∈ 𝑡 (𝑟 × {𝐴}) ⊆ (𝑓‘𝑟)))) → 𝑆 ∈ Top)
56		cmptop 22617	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Comp → 𝑅 ∈ Top)
57	52, 56	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑊 ∧ ∀𝑟 ∈ 𝑡 (𝑟 × {𝐴}) ⊆ (𝑓‘𝑟)))) → 𝑅 ∈ Top)
58		txtop 22791	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top)
59	57, 55, 58	syl2anc 584	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑊 ∧ ∀𝑟 ∈ 𝑡 (𝑟 × {𝐴}) ⊆ (𝑓‘𝑟)))) → (𝑅 ×t 𝑆) ∈ Top)
60		simprrl 778	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑊 ∧ ∀𝑟 ∈ 𝑡 (𝑟 × {𝐴}) ⊆ (𝑓‘𝑟)))) → 𝑓:𝑡⟶𝑊)
61	60	frnd 6643	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑊 ∧ ∀𝑟 ∈ 𝑡 (𝑟 × {𝐴}) ⊆ (𝑓‘𝑟)))) → ran 𝑓 ⊆ 𝑊)
62	11	ad2antrr 723	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑊 ∧ ∀𝑟 ∈ 𝑡 (𝑟 × {𝐴}) ⊆ (𝑓‘𝑟)))) → 𝑊 ⊆ (𝑅 ×t 𝑆))
63	61, 62	sstrd 3940	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑊 ∧ ∀𝑟 ∈ 𝑡 (𝑟 × {𝐴}) ⊆ (𝑓‘𝑟)))) → ran 𝑓 ⊆ (𝑅 ×t 𝑆))
64		uniopn 22117	. . . . . . . . 9 ⊢ (((𝑅 ×t 𝑆) ∈ Top ∧ ran 𝑓 ⊆ (𝑅 ×t 𝑆)) → ∪ ran 𝑓 ∈ (𝑅 ×t 𝑆))
65	59, 63, 64	syl2anc 584	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑊 ∧ ∀𝑟 ∈ 𝑡 (𝑟 × {𝐴}) ⊆ (𝑓‘𝑟)))) → ∪ ran 𝑓 ∈ (𝑅 ×t 𝑆))
66		simprrr 779	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑊 ∧ ∀𝑟 ∈ 𝑡 (𝑟 × {𝐴}) ⊆ (𝑓‘𝑟)))) → ∀𝑟 ∈ 𝑡 (𝑟 × {𝐴}) ⊆ (𝑓‘𝑟))
67		ss2iun 4953	. . . . . . . . . 10 ⊢ (∀𝑟 ∈ 𝑡 (𝑟 × {𝐴}) ⊆ (𝑓‘𝑟) → ∪ 𝑟 ∈ 𝑡 (𝑟 × {𝐴}) ⊆ ∪ 𝑟 ∈ 𝑡 (𝑓‘𝑟))
68	66, 67	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑊 ∧ ∀𝑟 ∈ 𝑡 (𝑟 × {𝐴}) ⊆ (𝑓‘𝑟)))) → ∪ 𝑟 ∈ 𝑡 (𝑟 × {𝐴}) ⊆ ∪ 𝑟 ∈ 𝑡 (𝑓‘𝑟))
69		simprl 768	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑊 ∧ ∀𝑟 ∈ 𝑡 (𝑟 × {𝐴}) ⊆ (𝑓‘𝑟)))) → 𝑋 = ∪ 𝑡)
70		uniiun 4999	. . . . . . . . . . . 12 ⊢ ∪ 𝑡 = ∪ 𝑟 ∈ 𝑡 𝑟
71	69, 70	eqtrdi 2793	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑊 ∧ ∀𝑟 ∈ 𝑡 (𝑟 × {𝐴}) ⊆ (𝑓‘𝑟)))) → 𝑋 = ∪ 𝑟 ∈ 𝑡 𝑟)
72	71	xpeq1d 5634	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑊 ∧ ∀𝑟 ∈ 𝑡 (𝑟 × {𝐴}) ⊆ (𝑓‘𝑟)))) → (𝑋 × {𝐴}) = (∪ 𝑟 ∈ 𝑡 𝑟 × {𝐴}))
73		xpiundir 5674	. . . . . . . . . 10 ⊢ (∪ 𝑟 ∈ 𝑡 𝑟 × {𝐴}) = ∪ 𝑟 ∈ 𝑡 (𝑟 × {𝐴})
74	72, 73	eqtr2di 2794	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑊 ∧ ∀𝑟 ∈ 𝑡 (𝑟 × {𝐴}) ⊆ (𝑓‘𝑟)))) → ∪ 𝑟 ∈ 𝑡 (𝑟 × {𝐴}) = (𝑋 × {𝐴}))
75	60	ffnd 6636	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑊 ∧ ∀𝑟 ∈ 𝑡 (𝑟 × {𝐴}) ⊆ (𝑓‘𝑟)))) → 𝑓 Fn 𝑡)
76		fniunfv 7157	. . . . . . . . . 10 ⊢ (𝑓 Fn 𝑡 → ∪ 𝑟 ∈ 𝑡 (𝑓‘𝑟) = ∪ ran 𝑓)
77	75, 76	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑊 ∧ ∀𝑟 ∈ 𝑡 (𝑟 × {𝐴}) ⊆ (𝑓‘𝑟)))) → ∪ 𝑟 ∈ 𝑡 (𝑓‘𝑟) = ∪ ran 𝑓)
78	68, 74, 77	3sstr3d 3976	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑊 ∧ ∀𝑟 ∈ 𝑡 (𝑟 × {𝐴}) ⊆ (𝑓‘𝑟)))) → (𝑋 × {𝐴}) ⊆ ∪ ran 𝑓)
79	3	ad2antrr 723	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑊 ∧ ∀𝑟 ∈ 𝑡 (𝑟 × {𝐴}) ⊆ (𝑓‘𝑟)))) → 𝐴 ∈ 𝑌)
80	47, 51, 52, 55, 65, 78, 79	txtube 22862	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑊 ∧ ∀𝑟 ∈ 𝑡 (𝑟 × {𝐴}) ⊆ (𝑓‘𝑟)))) → ∃𝑢 ∈ 𝑆 (𝐴 ∈ 𝑢 ∧ (𝑋 × 𝑢) ⊆ ∪ ran 𝑓))
81		vex 3445	. . . . . . . . . . . . . 14 ⊢ 𝑓 ∈ V
82	81	rnex 7802	. . . . . . . . . . . . 13 ⊢ ran 𝑓 ∈ V
83	82	elpw 4547	. . . . . . . . . . . 12 ⊢ (ran 𝑓 ∈ 𝒫 𝑊 ↔ ran 𝑓 ⊆ 𝑊)
84	61, 83	sylibr 233	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑊 ∧ ∀𝑟 ∈ 𝑡 (𝑟 × {𝐴}) ⊆ (𝑓‘𝑟)))) → ran 𝑓 ∈ 𝒫 𝑊)
85		simplr 766	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑊 ∧ ∀𝑟 ∈ 𝑡 (𝑟 × {𝐴}) ⊆ (𝑓‘𝑟)))) → 𝑡 ∈ (𝒫 𝑅 ∩ Fin))
86	85	elin2d 4143	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑊 ∧ ∀𝑟 ∈ 𝑡 (𝑟 × {𝐴}) ⊆ (𝑓‘𝑟)))) → 𝑡 ∈ Fin)
87		dffn4 6729	. . . . . . . . . . . . 13 ⊢ (𝑓 Fn 𝑡 ↔ 𝑓:𝑡–onto→ran 𝑓)
88	75, 87	sylib 217	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑊 ∧ ∀𝑟 ∈ 𝑡 (𝑟 × {𝐴}) ⊆ (𝑓‘𝑟)))) → 𝑓:𝑡–onto→ran 𝑓)
89		fofi 9173	. . . . . . . . . . . 12 ⊢ ((𝑡 ∈ Fin ∧ 𝑓:𝑡–onto→ran 𝑓) → ran 𝑓 ∈ Fin)
90	86, 88, 89	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑊 ∧ ∀𝑟 ∈ 𝑡 (𝑟 × {𝐴}) ⊆ (𝑓‘𝑟)))) → ran 𝑓 ∈ Fin)
91	84, 90	elind 4138	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑊 ∧ ∀𝑟 ∈ 𝑡 (𝑟 × {𝐴}) ⊆ (𝑓‘𝑟)))) → ran 𝑓 ∈ (𝒫 𝑊 ∩ Fin))
92		unieq 4859	. . . . . . . . . . . . 13 ⊢ (𝑣 = ran 𝑓 → ∪ 𝑣 = ∪ ran 𝑓)
93	92	sseq2d 3962	. . . . . . . . . . . 12 ⊢ (𝑣 = ran 𝑓 → ((𝑋 × 𝑢) ⊆ ∪ 𝑣 ↔ (𝑋 × 𝑢) ⊆ ∪ ran 𝑓))
94	93	rspcev 3570	. . . . . . . . . . 11 ⊢ ((ran 𝑓 ∈ (𝒫 𝑊 ∩ Fin) ∧ (𝑋 × 𝑢) ⊆ ∪ ran 𝑓) → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ ∪ 𝑣)
95	94	ex 413	. . . . . . . . . 10 ⊢ (ran 𝑓 ∈ (𝒫 𝑊 ∩ Fin) → ((𝑋 × 𝑢) ⊆ ∪ ran 𝑓 → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ ∪ 𝑣))
96	91, 95	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑊 ∧ ∀𝑟 ∈ 𝑡 (𝑟 × {𝐴}) ⊆ (𝑓‘𝑟)))) → ((𝑋 × 𝑢) ⊆ ∪ ran 𝑓 → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ ∪ 𝑣))
97	96	anim2d 612	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑊 ∧ ∀𝑟 ∈ 𝑡 (𝑟 × {𝐴}) ⊆ (𝑓‘𝑟)))) → ((𝐴 ∈ 𝑢 ∧ (𝑋 × 𝑢) ⊆ ∪ ran 𝑓) → (𝐴 ∈ 𝑢 ∧ ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ ∪ 𝑣)))
98	97	reximdv 3164	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑊 ∧ ∀𝑟 ∈ 𝑡 (𝑟 × {𝐴}) ⊆ (𝑓‘𝑟)))) → (∃𝑢 ∈ 𝑆 (𝐴 ∈ 𝑢 ∧ (𝑋 × 𝑢) ⊆ ∪ ran 𝑓) → ∃𝑢 ∈ 𝑆 (𝐴 ∈ 𝑢 ∧ ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ ∪ 𝑣)))
99	80, 98	mpd 15	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑊 ∧ ∀𝑟 ∈ 𝑡 (𝑟 × {𝐴}) ⊆ (𝑓‘𝑟)))) → ∃𝑢 ∈ 𝑆 (𝐴 ∈ 𝑢 ∧ ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ ∪ 𝑣))
100	99	expr 457	. . . . 5 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ 𝑋 = ∪ 𝑡) → ((𝑓:𝑡⟶𝑊 ∧ ∀𝑟 ∈ 𝑡 (𝑟 × {𝐴}) ⊆ (𝑓‘𝑟)) → ∃𝑢 ∈ 𝑆 (𝐴 ∈ 𝑢 ∧ ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ ∪ 𝑣)))
101	100	exlimdv 1935	. . . 4 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ 𝑋 = ∪ 𝑡) → (∃𝑓(𝑓:𝑡⟶𝑊 ∧ ∀𝑟 ∈ 𝑡 (𝑟 × {𝐴}) ⊆ (𝑓‘𝑟)) → ∃𝑢 ∈ 𝑆 (𝐴 ∈ 𝑢 ∧ ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ ∪ 𝑣)))
102	101	expimpd 454	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) → ((𝑋 = ∪ 𝑡 ∧ ∃𝑓(𝑓:𝑡⟶𝑊 ∧ ∀𝑟 ∈ 𝑡 (𝑟 × {𝐴}) ⊆ (𝑓‘𝑟))) → ∃𝑢 ∈ 𝑆 (𝐴 ∈ 𝑢 ∧ ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ ∪ 𝑣)))
103	102	rexlimdva 3149	. 2 ⊢ (𝜑 → (∃𝑡 ∈ (𝒫 𝑅 ∩ Fin)(𝑋 = ∪ 𝑡 ∧ ∃𝑓(𝑓:𝑡⟶𝑊 ∧ ∀𝑟 ∈ 𝑡 (𝑟 × {𝐴}) ⊆ (𝑓‘𝑟))) → ∃𝑢 ∈ 𝑆 (𝐴 ∈ 𝑢 ∧ ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ ∪ 𝑣)))
104	50, 103	mpd 15	1 ⊢ (𝜑 → ∃𝑢 ∈ 𝑆 (𝐴 ∈ 𝑢 ∧ ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ ∪ 𝑣))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1540  ∃wex 1780   ∈ wcel 2105  ∀wral 3062  ∃wrex 3071   ∩ cin 3895   ⊆ wss 3896  𝒫 cpw 4543  {csn 4569  ⟨cop 4575  ∪ cuni 4848  ∪ ciun 4935   × cxp 5603  ran crn 5606   Fn wfn 6458  ⟶wf 6459  –onto→wfo 6461  ‘cfv 6463  (class class class)co 7313  Fincfn 8779  Topctop 22113  Compccmp 22608   ×_t ctx 22782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-sep 5236  ax-nul 5243  ax-pow 5301  ax-pr 5365  ax-un 7626

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3351  df-rab 3405  df-v 3443  df-sbc 3726  df-csb 3842  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-pss 3915  df-nul 4267  df-if 4470  df-pw 4545  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4849  df-int 4891  df-iun 4937  df-iin 4938  df-br 5086  df-opab 5148  df-mpt 5169  df-tr 5203  df-id 5505  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5560  df-we 5562  df-xp 5611  df-rel 5612  df-cnv 5613  df-co 5614  df-dm 5615  df-rn 5616  df-res 5617  df-ima 5618  df-ord 6289  df-on 6290  df-lim 6291  df-suc 6292  df-iota 6415  df-fun 6465  df-fn 6466  df-f 6467  df-f1 6468  df-fo 6469  df-f1o 6470  df-fv 6471  df-ov 7316  df-oprab 7317  df-mpo 7318  df-om 7756  df-1st 7874  df-2nd 7875  df-1o 8342  df-er 8544  df-en 8780  df-dom 8781  df-fin 8783  df-topgen 17221  df-top 22114  df-bases 22167  df-cmp 22609  df-tx 22784

This theorem is referenced by:  txcmplem2  22864