Theorem txcmplem1 22252

Description: Lemma for txcmp 22254. (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 5595	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → ⟨𝑥, 𝐴⟩ ∈ (𝑋 × 𝑌))
5	2, 3, 4	syl2anr 598	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ⟨𝑥, 𝐴⟩ ∈ (𝑋 × 𝑌))
6		txcmp.u	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 × 𝑌) = ∪ 𝑊)
7	6	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑋 × 𝑌) = ∪ 𝑊)
8	5, 7	eleqtrd 2918	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ⟨𝑥, 𝐴⟩ ∈ ∪ 𝑊)
9		eluni2 4845	. . . . . . 7 ⊢ (⟨𝑥, 𝐴⟩ ∈ ∪ 𝑊 ↔ ∃𝑘 ∈ 𝑊 ⟨𝑥, 𝐴⟩ ∈ 𝑘)
10	8, 9	sylib 220	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∃𝑘 ∈ 𝑊 ⟨𝑥, 𝐴⟩ ∈ 𝑘)
11		txcmp.w	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑊 ⊆ (𝑅 ×t 𝑆))
12	11	adantr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑊 ⊆ (𝑅 ×t 𝑆))
13	12	sselda 3970	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝑊) → 𝑘 ∈ (𝑅 ×t 𝑆))
14		txcmp.s	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑆 ∈ Comp)
15		eltx 22179	. . . . . . . . . . . . 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 586	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑘 ∈ (𝑅 ×t 𝑆) ↔ ∀𝑦 ∈ 𝑘 ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (𝑦 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)))
17	16	adantr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑘 ∈ (𝑅 ×t 𝑆) ↔ ∀𝑦 ∈ 𝑘 ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (𝑦 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)))
18	17	biimpa 479	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (𝑅 ×t 𝑆)) → ∀𝑦 ∈ 𝑘 ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (𝑦 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘))
19	13, 18	syldan 593	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝑊) → ∀𝑦 ∈ 𝑘 ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (𝑦 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘))
20		eleq1 2903	. . . . . . . . . . . 12 ⊢ (𝑦 = ⟨𝑥, 𝐴⟩ → (𝑦 ∈ (𝑟 × 𝑠) ↔ ⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠)))
21	20	anbi1d 631	. . . . . . . . . . 11 ⊢ (𝑦 = ⟨𝑥, 𝐴⟩ → ((𝑦 ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘) ↔ (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)))
22	21	2rexbidv 3303	. . . . . . . . . 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 3623	. . . . . . . . 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 5599	. . . . . . . . . . . . 13 ⊢ (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) → 𝑥 ∈ 𝑟)
26	25	ad2antrl 726	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝑊) ∧ (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)) → 𝑥 ∈ 𝑟)
27		opelxp2 5600	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) → 𝐴 ∈ 𝑠)
28	27	ad2antrl 726	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝑊) ∧ (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)) → 𝐴 ∈ 𝑠)
29	28	snssd 4745	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝑊) ∧ (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)) → {𝐴} ⊆ 𝑠)
30		xpss2 5578	. . . . . . . . . . . . . 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 771	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝑊) ∧ (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)) → (𝑟 × 𝑠) ⊆ 𝑘)
33	31, 32	sstrd 3980	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝑊) ∧ (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)) → (𝑟 × {𝐴}) ⊆ 𝑘)
34	26, 33	jca 514	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝑊) ∧ (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘)) → (𝑥 ∈ 𝑟 ∧ (𝑟 × {𝐴}) ⊆ 𝑘))
35	34	ex 415	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝑊) → ((⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘) → (𝑥 ∈ 𝑟 ∧ (𝑟 × {𝐴}) ⊆ 𝑘)))
36	35	rexlimdvw 3293	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝑊) → (∃𝑠 ∈ 𝑆 (⟨𝑥, 𝐴⟩ ∈ (𝑟 × 𝑠) ∧ (𝑟 × 𝑠) ⊆ 𝑘) → (𝑥 ∈ 𝑟 ∧ (𝑟 × {𝐴}) ⊆ 𝑘)))
37	36	reximdv 3276	. . . . . . . 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 3277	. . . . . 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 3358	. . . . . 6 ⊢ (∃𝑘 ∈ 𝑊 ∃𝑟 ∈ 𝑅 (𝑥 ∈ 𝑟 ∧ (𝑟 × {𝐴}) ⊆ 𝑘) ↔ ∃𝑟 ∈ 𝑅 ∃𝑘 ∈ 𝑊 (𝑥 ∈ 𝑟 ∧ (𝑟 × {𝐴}) ⊆ 𝑘))
42		r19.42v 3353	. . . . . . 7 ⊢ (∃𝑘 ∈ 𝑊 (𝑥 ∈ 𝑟 ∧ (𝑟 × {𝐴}) ⊆ 𝑘) ↔ (𝑥 ∈ 𝑟 ∧ ∃𝑘 ∈ 𝑊 (𝑟 × {𝐴}) ⊆ 𝑘))
43	42	rexbii 3250	. . . . . 6 ⊢ (∃𝑟 ∈ 𝑅 ∃𝑘 ∈ 𝑊 (𝑥 ∈ 𝑟 ∧ (𝑟 × {𝐴}) ⊆ 𝑘) ↔ ∃𝑟 ∈ 𝑅 (𝑥 ∈ 𝑟 ∧ ∃𝑘 ∈ 𝑊 (𝑟 × {𝐴}) ⊆ 𝑘))
44	41, 43	bitri 277	. . . . 5 ⊢ (∃𝑘 ∈ 𝑊 ∃𝑟 ∈ 𝑅 (𝑥 ∈ 𝑟 ∧ (𝑟 × {𝐴}) ⊆ 𝑘) ↔ ∃𝑟 ∈ 𝑅 (𝑥 ∈ 𝑟 ∧ ∃𝑘 ∈ 𝑊 (𝑟 × {𝐴}) ⊆ 𝑘))
45	40, 44	sylib 220	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∃𝑟 ∈ 𝑅 (𝑥 ∈ 𝑟 ∧ ∃𝑘 ∈ 𝑊 (𝑟 × {𝐴}) ⊆ 𝑘))
46	45	ralrimiva 3185	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∃𝑟 ∈ 𝑅 (𝑥 ∈ 𝑟 ∧ ∃𝑘 ∈ 𝑊 (𝑟 × {𝐴}) ⊆ 𝑘))
47		txcmp.x	. . . 4 ⊢ 𝑋 = ∪ 𝑅
48		sseq2 3996	. . . 4 ⊢ (𝑘 = (𝑓‘𝑟) → ((𝑟 × {𝐴}) ⊆ 𝑘 ↔ (𝑟 × {𝐴}) ⊆ (𝑓‘𝑟)))
49	47, 48	cmpcovf 22002	. . 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 586	. 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 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑊 ∧ ∀𝑟 ∈ 𝑡 (𝑟 × {𝐴}) ⊆ (𝑓‘𝑟)))) → 𝑅 ∈ Comp)
53		cmptop 22006	. . . . . . . . . 10 ⊢ (𝑆 ∈ Comp → 𝑆 ∈ Top)
54	14, 53	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ Top)
55	54	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑊 ∧ ∀𝑟 ∈ 𝑡 (𝑟 × {𝐴}) ⊆ (𝑓‘𝑟)))) → 𝑆 ∈ Top)
56		cmptop 22006	. . . . . . . . . . 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 22180	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top)
59	57, 55, 58	syl2anc 586	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑊 ∧ ∀𝑟 ∈ 𝑡 (𝑟 × {𝐴}) ⊆ (𝑓‘𝑟)))) → (𝑅 ×t 𝑆) ∈ Top)
60		simprrl 779	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑊 ∧ ∀𝑟 ∈ 𝑡 (𝑟 × {𝐴}) ⊆ (𝑓‘𝑟)))) → 𝑓:𝑡⟶𝑊)
61	60	frnd 6524	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑊 ∧ ∀𝑟 ∈ 𝑡 (𝑟 × {𝐴}) ⊆ (𝑓‘𝑟)))) → ran 𝑓 ⊆ 𝑊)
62	11	ad2antrr 724	. . . . . . . . . 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 3980	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑊 ∧ ∀𝑟 ∈ 𝑡 (𝑟 × {𝐴}) ⊆ (𝑓‘𝑟)))) → ran 𝑓 ⊆ (𝑅 ×t 𝑆))
64		uniopn 21508	. . . . . . . . 9 ⊢ (((𝑅 ×t 𝑆) ∈ Top ∧ ran 𝑓 ⊆ (𝑅 ×t 𝑆)) → ∪ ran 𝑓 ∈ (𝑅 ×t 𝑆))
65	59, 63, 64	syl2anc 586	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑊 ∧ ∀𝑟 ∈ 𝑡 (𝑟 × {𝐴}) ⊆ (𝑓‘𝑟)))) → ∪ ran 𝑓 ∈ (𝑅 ×t 𝑆))
66		simprrr 780	. . . . . . . . . 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 4940	. . . . . . . . . 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 769	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑊 ∧ ∀𝑟 ∈ 𝑡 (𝑟 × {𝐴}) ⊆ (𝑓‘𝑟)))) → 𝑋 = ∪ 𝑡)
70		uniiun 4985	. . . . . . . . . . . 12 ⊢ ∪ 𝑡 = ∪ 𝑟 ∈ 𝑡 𝑟
71	69, 70	syl6eq 2875	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑊 ∧ ∀𝑟 ∈ 𝑡 (𝑟 × {𝐴}) ⊆ (𝑓‘𝑟)))) → 𝑋 = ∪ 𝑟 ∈ 𝑡 𝑟)
72	71	xpeq1d 5587	. . . . . . . . . 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 5626	. . . . . . . . . 10 ⊢ (∪ 𝑟 ∈ 𝑡 𝑟 × {𝐴}) = ∪ 𝑟 ∈ 𝑡 (𝑟 × {𝐴})
74	72, 73	syl6req 2876	. . . . . . . . 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 6518	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑊 ∧ ∀𝑟 ∈ 𝑡 (𝑟 × {𝐴}) ⊆ (𝑓‘𝑟)))) → 𝑓 Fn 𝑡)
76		fniunfv 7009	. . . . . . . . . 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 4016	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑊 ∧ ∀𝑟 ∈ 𝑡 (𝑟 × {𝐴}) ⊆ (𝑓‘𝑟)))) → (𝑋 × {𝐴}) ⊆ ∪ ran 𝑓)
79	3	ad2antrr 724	. . . . . . . 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 22251	. . . . . . 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 3500	. . . . . . . . . . . . . 14 ⊢ 𝑓 ∈ V
82	81	rnex 7620	. . . . . . . . . . . . 13 ⊢ ran 𝑓 ∈ V
83	82	elpw 4546	. . . . . . . . . . . 12 ⊢ (ran 𝑓 ∈ 𝒫 𝑊 ↔ ran 𝑓 ⊆ 𝑊)
84	61, 83	sylibr 236	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑊 ∧ ∀𝑟 ∈ 𝑡 (𝑟 × {𝐴}) ⊆ (𝑓‘𝑟)))) → ran 𝑓 ∈ 𝒫 𝑊)
85		simplr 767	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑊 ∧ ∀𝑟 ∈ 𝑡 (𝑟 × {𝐴}) ⊆ (𝑓‘𝑟)))) → 𝑡 ∈ (𝒫 𝑅 ∩ Fin))
86	85	elin2d 4179	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑊 ∧ ∀𝑟 ∈ 𝑡 (𝑟 × {𝐴}) ⊆ (𝑓‘𝑟)))) → 𝑡 ∈ Fin)
87		dffn4 6599	. . . . . . . . . . . . 13 ⊢ (𝑓 Fn 𝑡 ↔ 𝑓:𝑡–onto→ran 𝑓)
88	75, 87	sylib 220	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑊 ∧ ∀𝑟 ∈ 𝑡 (𝑟 × {𝐴}) ⊆ (𝑓‘𝑟)))) → 𝑓:𝑡–onto→ran 𝑓)
89		fofi 8813	. . . . . . . . . . . 12 ⊢ ((𝑡 ∈ Fin ∧ 𝑓:𝑡–onto→ran 𝑓) → ran 𝑓 ∈ Fin)
90	86, 88, 89	syl2anc 586	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑊 ∧ ∀𝑟 ∈ 𝑡 (𝑟 × {𝐴}) ⊆ (𝑓‘𝑟)))) → ran 𝑓 ∈ Fin)
91	84, 90	elind 4174	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝒫 𝑅 ∩ Fin)) ∧ (𝑋 = ∪ 𝑡 ∧ (𝑓:𝑡⟶𝑊 ∧ ∀𝑟 ∈ 𝑡 (𝑟 × {𝐴}) ⊆ (𝑓‘𝑟)))) → ran 𝑓 ∈ (𝒫 𝑊 ∩ Fin))
92		unieq 4852	. . . . . . . . . . . . 13 ⊢ (𝑣 = ran 𝑓 → ∪ 𝑣 = ∪ ran 𝑓)
93	92	sseq2d 4002	. . . . . . . . . . . 12 ⊢ (𝑣 = ran 𝑓 → ((𝑋 × 𝑢) ⊆ ∪ 𝑣 ↔ (𝑋 × 𝑢) ⊆ ∪ ran 𝑓))
94	93	rspcev 3626	. . . . . . . . . . 11 ⊢ ((ran 𝑓 ∈ (𝒫 𝑊 ∩ Fin) ∧ (𝑋 × 𝑢) ⊆ ∪ ran 𝑓) → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ ∪ 𝑣)
95	94	ex 415	. . . . . . . . . 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 613	. . . . . . . 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 3276	. . . . . . 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 459	. . . . 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 1933	. . . 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 456	. . 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 3287	. 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 208   ∧ wa 398   = wceq 1536  ∃wex 1779   ∈ wcel 2113  ∀wral 3141  ∃wrex 3142   ∩ cin 3938   ⊆ wss 3939  𝒫 cpw 4542  {csn 4570  ⟨cop 4576  ∪ cuni 4841  ∪ ciun 4922   × cxp 5556  ran crn 5559   Fn wfn 6353  ⟶wf 6354  –onto→wfo 6356  ‘cfv 6358  (class class class)co 7159  Fincfn 8512  Topctop 21504  Compccmp 21997   ×_t ctx 22171

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-reu 3148  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4842  df-int 4880  df-iun 4924  df-iin 4925  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-we 5519  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-pred 6151  df-ord 6197  df-on 6198  df-lim 6199  df-suc 6200  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-ov 7162  df-oprab 7163  df-mpo 7164  df-om 7584  df-1st 7692  df-2nd 7693  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-1o 8105  df-oadd 8109  df-er 8292  df-en 8513  df-dom 8514  df-fin 8516  df-topgen 16720  df-top 21505  df-bases 21557  df-cmp 21998  df-tx 22173

This theorem is referenced by:  txcmplem2  22253