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Theorem txflf 23921

Description: Two sequences converge in a filter iff the sequence of their ordered pairs converges. (Contributed by Mario Carneiro, 19-Sep-2015.)

**Hypotheses**
Ref	Expression
txflf.j	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
txflf.k	⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
txflf.l	⊢ (𝜑 → 𝐿 ∈ (Fil‘𝑍))
txflf.f	⊢ (𝜑 → 𝐹:𝑍⟶𝑋)
txflf.g	⊢ (𝜑 → 𝐺:𝑍⟶𝑌)
txflf.h	⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ ⟨(𝐹‘𝑛), (𝐺‘𝑛)⟩)

**Assertion**
Ref	Expression
txflf	⊢ (𝜑 → (⟨𝑅, 𝑆⟩ ∈ (((𝐽 ×_t 𝐾) fLimf 𝐿)‘𝐻) ↔ (𝑅 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝑆 ∈ ((𝐾 fLimf 𝐿)‘𝐺))))

Distinct variable groups: 𝜑,𝑛 𝑛,𝐹 𝑛,𝐺 𝑛,𝑍 𝑛,𝑋 𝑛,𝑌 Allowed substitution hints: 𝑅(𝑛) 𝑆(𝑛) 𝐻(𝑛) 𝐽(𝑛) 𝐾(𝑛) 𝐿(𝑛)

Proof of Theorem txflf Dummy variables 𝑢 ℎ 𝑣 𝑧 𝑓 𝑔 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3440	. . . . . . . 8 ⊢ 𝑢 ∈ V
2		vex 3440	. . . . . . . 8 ⊢ 𝑣 ∈ V
3	1, 2	xpex 7686	. . . . . . 7 ⊢ (𝑢 × 𝑣) ∈ V
4	3	rgen2w 3052	. . . . . 6 ⊢ ∀𝑢 ∈ 𝐽 ∀𝑣 ∈ 𝐾 (𝑢 × 𝑣) ∈ V
5		eqid 2731	. . . . . . 7 ⊢ (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣)) = (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣))
6		eleq2 2820	. . . . . . . 8 ⊢ (𝑧 = (𝑢 × 𝑣) → (⟨𝑅, 𝑆⟩ ∈ 𝑧 ↔ ⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣)))
7		sseq2 3956	. . . . . . . . 9 ⊢ (𝑧 = (𝑢 × 𝑣) → ((𝐻 “ ℎ) ⊆ 𝑧 ↔ (𝐻 “ ℎ) ⊆ (𝑢 × 𝑣)))
8	7	rexbidv 3156	. . . . . . . 8 ⊢ (𝑧 = (𝑢 × 𝑣) → (∃ℎ ∈ 𝐿 (𝐻 “ ℎ) ⊆ 𝑧 ↔ ∃ℎ ∈ 𝐿 (𝐻 “ ℎ) ⊆ (𝑢 × 𝑣)))
9	6, 8	imbi12d 344	. . . . . . 7 ⊢ (𝑧 = (𝑢 × 𝑣) → ((⟨𝑅, 𝑆⟩ ∈ 𝑧 → ∃ℎ ∈ 𝐿 (𝐻 “ ℎ) ⊆ 𝑧) ↔ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) → ∃ℎ ∈ 𝐿 (𝐻 “ ℎ) ⊆ (𝑢 × 𝑣))))
10	5, 9	ralrnmpo 7485	. . . . . 6 ⊢ (∀𝑢 ∈ 𝐽 ∀𝑣 ∈ 𝐾 (𝑢 × 𝑣) ∈ V → (∀𝑧 ∈ ran (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣))(⟨𝑅, 𝑆⟩ ∈ 𝑧 → ∃ℎ ∈ 𝐿 (𝐻 “ ℎ) ⊆ 𝑧) ↔ ∀𝑢 ∈ 𝐽 ∀𝑣 ∈ 𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) → ∃ℎ ∈ 𝐿 (𝐻 “ ℎ) ⊆ (𝑢 × 𝑣))))
11	4, 10	ax-mp 5	. . . . 5 ⊢ (∀𝑧 ∈ ran (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣))(⟨𝑅, 𝑆⟩ ∈ 𝑧 → ∃ℎ ∈ 𝐿 (𝐻 “ ℎ) ⊆ 𝑧) ↔ ∀𝑢 ∈ 𝐽 ∀𝑣 ∈ 𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) → ∃ℎ ∈ 𝐿 (𝐻 “ ℎ) ⊆ (𝑢 × 𝑣)))
12		opelxp 5650	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ↔ (𝑅 ∈ 𝑢 ∧ 𝑆 ∈ 𝑣))
13	12	biancomi 462	. . . . . . . . . . . . . . 15 ⊢ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ↔ (𝑆 ∈ 𝑣 ∧ 𝑅 ∈ 𝑢))
14	13	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ↔ (𝑆 ∈ 𝑣 ∧ 𝑅 ∈ 𝑢)))
15		r19.40 3098	. . . . . . . . . . . . . . . . 17 ⊢ (∃ℎ ∈ 𝐿 (∀𝑛 ∈ ℎ (𝐹‘𝑛) ∈ 𝑢 ∧ ∀𝑛 ∈ ℎ (𝐺‘𝑛) ∈ 𝑣) → (∃ℎ ∈ 𝐿 ∀𝑛 ∈ ℎ (𝐹‘𝑛) ∈ 𝑢 ∧ ∃ℎ ∈ 𝐿 ∀𝑛 ∈ ℎ (𝐺‘𝑛) ∈ 𝑣))
16		raleq 3289	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℎ = 𝑓 → (∀𝑛 ∈ ℎ (𝐹‘𝑛) ∈ 𝑢 ↔ ∀𝑛 ∈ 𝑓 (𝐹‘𝑛) ∈ 𝑢))
17	16	cbvrexvw 3211	. . . . . . . . . . . . . . . . . 18 ⊢ (∃ℎ ∈ 𝐿 ∀𝑛 ∈ ℎ (𝐹‘𝑛) ∈ 𝑢 ↔ ∃𝑓 ∈ 𝐿 ∀𝑛 ∈ 𝑓 (𝐹‘𝑛) ∈ 𝑢)
18		raleq 3289	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℎ = 𝑔 → (∀𝑛 ∈ ℎ (𝐺‘𝑛) ∈ 𝑣 ↔ ∀𝑛 ∈ 𝑔 (𝐺‘𝑛) ∈ 𝑣))
19	18	cbvrexvw 3211	. . . . . . . . . . . . . . . . . 18 ⊢ (∃ℎ ∈ 𝐿 ∀𝑛 ∈ ℎ (𝐺‘𝑛) ∈ 𝑣 ↔ ∃𝑔 ∈ 𝐿 ∀𝑛 ∈ 𝑔 (𝐺‘𝑛) ∈ 𝑣)
20	17, 19	anbi12i 628	. . . . . . . . . . . . . . . . 17 ⊢ ((∃ℎ ∈ 𝐿 ∀𝑛 ∈ ℎ (𝐹‘𝑛) ∈ 𝑢 ∧ ∃ℎ ∈ 𝐿 ∀𝑛 ∈ ℎ (𝐺‘𝑛) ∈ 𝑣) ↔ (∃𝑓 ∈ 𝐿 ∀𝑛 ∈ 𝑓 (𝐹‘𝑛) ∈ 𝑢 ∧ ∃𝑔 ∈ 𝐿 ∀𝑛 ∈ 𝑔 (𝐺‘𝑛) ∈ 𝑣))
21	15, 20	sylib 218	. . . . . . . . . . . . . . . 16 ⊢ (∃ℎ ∈ 𝐿 (∀𝑛 ∈ ℎ (𝐹‘𝑛) ∈ 𝑢 ∧ ∀𝑛 ∈ ℎ (𝐺‘𝑛) ∈ 𝑣) → (∃𝑓 ∈ 𝐿 ∀𝑛 ∈ 𝑓 (𝐹‘𝑛) ∈ 𝑢 ∧ ∃𝑔 ∈ 𝐿 ∀𝑛 ∈ 𝑔 (𝐺‘𝑛) ∈ 𝑣))
22		reeanv 3204	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑓 ∈ 𝐿 ∃𝑔 ∈ 𝐿 (∀𝑛 ∈ 𝑓 (𝐹‘𝑛) ∈ 𝑢 ∧ ∀𝑛 ∈ 𝑔 (𝐺‘𝑛) ∈ 𝑣) ↔ (∃𝑓 ∈ 𝐿 ∀𝑛 ∈ 𝑓 (𝐹‘𝑛) ∈ 𝑢 ∧ ∃𝑔 ∈ 𝐿 ∀𝑛 ∈ 𝑔 (𝐺‘𝑛) ∈ 𝑣))
23		txflf.l	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐿 ∈ (Fil‘𝑍))
24		filin 23769	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐿 ∈ (Fil‘𝑍) ∧ 𝑓 ∈ 𝐿 ∧ 𝑔 ∈ 𝐿) → (𝑓 ∩ 𝑔) ∈ 𝐿)
25	24	3expb 1120	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐿 ∈ (Fil‘𝑍) ∧ (𝑓 ∈ 𝐿 ∧ 𝑔 ∈ 𝐿)) → (𝑓 ∩ 𝑔) ∈ 𝐿)
26	23, 25	sylan 580	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐿 ∧ 𝑔 ∈ 𝐿)) → (𝑓 ∩ 𝑔) ∈ 𝐿)
27		inss1 4184	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓 ∩ 𝑔) ⊆ 𝑓
28		ssralv 3998	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑓 ∩ 𝑔) ⊆ 𝑓 → (∀𝑛 ∈ 𝑓 (𝐹‘𝑛) ∈ 𝑢 → ∀𝑛 ∈ (𝑓 ∩ 𝑔)(𝐹‘𝑛) ∈ 𝑢))
29	27, 28	ax-mp 5	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∀𝑛 ∈ 𝑓 (𝐹‘𝑛) ∈ 𝑢 → ∀𝑛 ∈ (𝑓 ∩ 𝑔)(𝐹‘𝑛) ∈ 𝑢)
30		inss2 4185	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓 ∩ 𝑔) ⊆ 𝑔
31		ssralv 3998	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑓 ∩ 𝑔) ⊆ 𝑔 → (∀𝑛 ∈ 𝑔 (𝐺‘𝑛) ∈ 𝑣 → ∀𝑛 ∈ (𝑓 ∩ 𝑔)(𝐺‘𝑛) ∈ 𝑣))
32	30, 31	ax-mp 5	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∀𝑛 ∈ 𝑔 (𝐺‘𝑛) ∈ 𝑣 → ∀𝑛 ∈ (𝑓 ∩ 𝑔)(𝐺‘𝑛) ∈ 𝑣)
33	29, 32	anim12i 613	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((∀𝑛 ∈ 𝑓 (𝐹‘𝑛) ∈ 𝑢 ∧ ∀𝑛 ∈ 𝑔 (𝐺‘𝑛) ∈ 𝑣) → (∀𝑛 ∈ (𝑓 ∩ 𝑔)(𝐹‘𝑛) ∈ 𝑢 ∧ ∀𝑛 ∈ (𝑓 ∩ 𝑔)(𝐺‘𝑛) ∈ 𝑣))
34		raleq 3289	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℎ = (𝑓 ∩ 𝑔) → (∀𝑛 ∈ ℎ (𝐹‘𝑛) ∈ 𝑢 ↔ ∀𝑛 ∈ (𝑓 ∩ 𝑔)(𝐹‘𝑛) ∈ 𝑢))
35		raleq 3289	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℎ = (𝑓 ∩ 𝑔) → (∀𝑛 ∈ ℎ (𝐺‘𝑛) ∈ 𝑣 ↔ ∀𝑛 ∈ (𝑓 ∩ 𝑔)(𝐺‘𝑛) ∈ 𝑣))
36	34, 35	anbi12d 632	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℎ = (𝑓 ∩ 𝑔) → ((∀𝑛 ∈ ℎ (𝐹‘𝑛) ∈ 𝑢 ∧ ∀𝑛 ∈ ℎ (𝐺‘𝑛) ∈ 𝑣) ↔ (∀𝑛 ∈ (𝑓 ∩ 𝑔)(𝐹‘𝑛) ∈ 𝑢 ∧ ∀𝑛 ∈ (𝑓 ∩ 𝑔)(𝐺‘𝑛) ∈ 𝑣)))
37	36	rspcev 3572	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑓 ∩ 𝑔) ∈ 𝐿 ∧ (∀𝑛 ∈ (𝑓 ∩ 𝑔)(𝐹‘𝑛) ∈ 𝑢 ∧ ∀𝑛 ∈ (𝑓 ∩ 𝑔)(𝐺‘𝑛) ∈ 𝑣)) → ∃ℎ ∈ 𝐿 (∀𝑛 ∈ ℎ (𝐹‘𝑛) ∈ 𝑢 ∧ ∀𝑛 ∈ ℎ (𝐺‘𝑛) ∈ 𝑣))
38	26, 33, 37	syl2an 596	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑓 ∈ 𝐿 ∧ 𝑔 ∈ 𝐿)) ∧ (∀𝑛 ∈ 𝑓 (𝐹‘𝑛) ∈ 𝑢 ∧ ∀𝑛 ∈ 𝑔 (𝐺‘𝑛) ∈ 𝑣)) → ∃ℎ ∈ 𝐿 (∀𝑛 ∈ ℎ (𝐹‘𝑛) ∈ 𝑢 ∧ ∀𝑛 ∈ ℎ (𝐺‘𝑛) ∈ 𝑣))
39	38	ex 412	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐿 ∧ 𝑔 ∈ 𝐿)) → ((∀𝑛 ∈ 𝑓 (𝐹‘𝑛) ∈ 𝑢 ∧ ∀𝑛 ∈ 𝑔 (𝐺‘𝑛) ∈ 𝑣) → ∃ℎ ∈ 𝐿 (∀𝑛 ∈ ℎ (𝐹‘𝑛) ∈ 𝑢 ∧ ∀𝑛 ∈ ℎ (𝐺‘𝑛) ∈ 𝑣)))
40	39	rexlimdvva 3189	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (∃𝑓 ∈ 𝐿 ∃𝑔 ∈ 𝐿 (∀𝑛 ∈ 𝑓 (𝐹‘𝑛) ∈ 𝑢 ∧ ∀𝑛 ∈ 𝑔 (𝐺‘𝑛) ∈ 𝑣) → ∃ℎ ∈ 𝐿 (∀𝑛 ∈ ℎ (𝐹‘𝑛) ∈ 𝑢 ∧ ∀𝑛 ∈ ℎ (𝐺‘𝑛) ∈ 𝑣)))
41	22, 40	biimtrrid 243	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((∃𝑓 ∈ 𝐿 ∀𝑛 ∈ 𝑓 (𝐹‘𝑛) ∈ 𝑢 ∧ ∃𝑔 ∈ 𝐿 ∀𝑛 ∈ 𝑔 (𝐺‘𝑛) ∈ 𝑣) → ∃ℎ ∈ 𝐿 (∀𝑛 ∈ ℎ (𝐹‘𝑛) ∈ 𝑢 ∧ ∀𝑛 ∈ ℎ (𝐺‘𝑛) ∈ 𝑣)))
42	21, 41	impbid2 226	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (∃ℎ ∈ 𝐿 (∀𝑛 ∈ ℎ (𝐹‘𝑛) ∈ 𝑢 ∧ ∀𝑛 ∈ ℎ (𝐺‘𝑛) ∈ 𝑣) ↔ (∃𝑓 ∈ 𝐿 ∀𝑛 ∈ 𝑓 (𝐹‘𝑛) ∈ 𝑢 ∧ ∃𝑔 ∈ 𝐿 ∀𝑛 ∈ 𝑔 (𝐺‘𝑛) ∈ 𝑣)))
43		df-ima 5627	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐻 “ ℎ) = ran (𝐻 ↾ ℎ)
44		filelss 23767	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐿 ∈ (Fil‘𝑍) ∧ ℎ ∈ 𝐿) → ℎ ⊆ 𝑍)
45	23, 44	sylan 580	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ ℎ ∈ 𝐿) → ℎ ⊆ 𝑍)
46		txflf.h	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ ⟨(𝐹‘𝑛), (𝐺‘𝑛)⟩)
47	46	reseq1i 5923	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐻 ↾ ℎ) = ((𝑛 ∈ 𝑍 ↦ ⟨(𝐹‘𝑛), (𝐺‘𝑛)⟩) ↾ ℎ)
48		resmpt 5985	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℎ ⊆ 𝑍 → ((𝑛 ∈ 𝑍 ↦ ⟨(𝐹‘𝑛), (𝐺‘𝑛)⟩) ↾ ℎ) = (𝑛 ∈ ℎ ↦ ⟨(𝐹‘𝑛), (𝐺‘𝑛)⟩))
49	47, 48	eqtrid 2778	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℎ ⊆ 𝑍 → (𝐻 ↾ ℎ) = (𝑛 ∈ ℎ ↦ ⟨(𝐹‘𝑛), (𝐺‘𝑛)⟩))
50	45, 49	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ ℎ ∈ 𝐿) → (𝐻 ↾ ℎ) = (𝑛 ∈ ℎ ↦ ⟨(𝐹‘𝑛), (𝐺‘𝑛)⟩))
51	50	rneqd 5877	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ ℎ ∈ 𝐿) → ran (𝐻 ↾ ℎ) = ran (𝑛 ∈ ℎ ↦ ⟨(𝐹‘𝑛), (𝐺‘𝑛)⟩))
52	43, 51	eqtrid 2778	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ℎ ∈ 𝐿) → (𝐻 “ ℎ) = ran (𝑛 ∈ ℎ ↦ ⟨(𝐹‘𝑛), (𝐺‘𝑛)⟩))
53	52	sseq1d 3961	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ℎ ∈ 𝐿) → ((𝐻 “ ℎ) ⊆ (𝑢 × 𝑣) ↔ ran (𝑛 ∈ ℎ ↦ ⟨(𝐹‘𝑛), (𝐺‘𝑛)⟩) ⊆ (𝑢 × 𝑣)))
54		opelxp 5650	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟨(𝐹‘𝑛), (𝐺‘𝑛)⟩ ∈ (𝑢 × 𝑣) ↔ ((𝐹‘𝑛) ∈ 𝑢 ∧ (𝐺‘𝑛) ∈ 𝑣))
55	54	ralbii 3078	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑛 ∈ ℎ ⟨(𝐹‘𝑛), (𝐺‘𝑛)⟩ ∈ (𝑢 × 𝑣) ↔ ∀𝑛 ∈ ℎ ((𝐹‘𝑛) ∈ 𝑢 ∧ (𝐺‘𝑛) ∈ 𝑣))
56		eqid 2731	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ℎ ↦ ⟨(𝐹‘𝑛), (𝐺‘𝑛)⟩) = (𝑛 ∈ ℎ ↦ ⟨(𝐹‘𝑛), (𝐺‘𝑛)⟩)
57	56	fmpt 7043	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑛 ∈ ℎ ⟨(𝐹‘𝑛), (𝐺‘𝑛)⟩ ∈ (𝑢 × 𝑣) ↔ (𝑛 ∈ ℎ ↦ ⟨(𝐹‘𝑛), (𝐺‘𝑛)⟩):ℎ⟶(𝑢 × 𝑣))
58		opex 5402	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ⟨(𝐹‘𝑛), (𝐺‘𝑛)⟩ ∈ V
59	58, 56	fnmpti 6624	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ℎ ↦ ⟨(𝐹‘𝑛), (𝐺‘𝑛)⟩) Fn ℎ
60		df-f 6485	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 ∈ ℎ ↦ ⟨(𝐹‘𝑛), (𝐺‘𝑛)⟩):ℎ⟶(𝑢 × 𝑣) ↔ ((𝑛 ∈ ℎ ↦ ⟨(𝐹‘𝑛), (𝐺‘𝑛)⟩) Fn ℎ ∧ ran (𝑛 ∈ ℎ ↦ ⟨(𝐹‘𝑛), (𝐺‘𝑛)⟩) ⊆ (𝑢 × 𝑣)))
61	59, 60	mpbiran 709	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 ∈ ℎ ↦ ⟨(𝐹‘𝑛), (𝐺‘𝑛)⟩):ℎ⟶(𝑢 × 𝑣) ↔ ran (𝑛 ∈ ℎ ↦ ⟨(𝐹‘𝑛), (𝐺‘𝑛)⟩) ⊆ (𝑢 × 𝑣))
62	57, 61	bitri 275	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑛 ∈ ℎ ⟨(𝐹‘𝑛), (𝐺‘𝑛)⟩ ∈ (𝑢 × 𝑣) ↔ ran (𝑛 ∈ ℎ ↦ ⟨(𝐹‘𝑛), (𝐺‘𝑛)⟩) ⊆ (𝑢 × 𝑣))
63		r19.26 3092	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑛 ∈ ℎ ((𝐹‘𝑛) ∈ 𝑢 ∧ (𝐺‘𝑛) ∈ 𝑣) ↔ (∀𝑛 ∈ ℎ (𝐹‘𝑛) ∈ 𝑢 ∧ ∀𝑛 ∈ ℎ (𝐺‘𝑛) ∈ 𝑣))
64	55, 62, 63	3bitr3i 301	. . . . . . . . . . . . . . . . 17 ⊢ (ran (𝑛 ∈ ℎ ↦ ⟨(𝐹‘𝑛), (𝐺‘𝑛)⟩) ⊆ (𝑢 × 𝑣) ↔ (∀𝑛 ∈ ℎ (𝐹‘𝑛) ∈ 𝑢 ∧ ∀𝑛 ∈ ℎ (𝐺‘𝑛) ∈ 𝑣))
65	53, 64	bitrdi 287	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ℎ ∈ 𝐿) → ((𝐻 “ ℎ) ⊆ (𝑢 × 𝑣) ↔ (∀𝑛 ∈ ℎ (𝐹‘𝑛) ∈ 𝑢 ∧ ∀𝑛 ∈ ℎ (𝐺‘𝑛) ∈ 𝑣)))
66	65	rexbidva 3154	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (∃ℎ ∈ 𝐿 (𝐻 “ ℎ) ⊆ (𝑢 × 𝑣) ↔ ∃ℎ ∈ 𝐿 (∀𝑛 ∈ ℎ (𝐹‘𝑛) ∈ 𝑢 ∧ ∀𝑛 ∈ ℎ (𝐺‘𝑛) ∈ 𝑣)))
67		txflf.f	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐹:𝑍⟶𝑋)
68	67	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐿) → 𝐹:𝑍⟶𝑋)
69	68	ffund 6655	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐿) → Fun 𝐹)
70		filelss 23767	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐿 ∈ (Fil‘𝑍) ∧ 𝑓 ∈ 𝐿) → 𝑓 ⊆ 𝑍)
71	23, 70	sylan 580	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐿) → 𝑓 ⊆ 𝑍)
72	68	fdmd 6661	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐿) → dom 𝐹 = 𝑍)
73	71, 72	sseqtrrd 3967	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐿) → 𝑓 ⊆ dom 𝐹)
74		funimass4 6886	. . . . . . . . . . . . . . . . . 18 ⊢ ((Fun 𝐹 ∧ 𝑓 ⊆ dom 𝐹) → ((𝐹 “ 𝑓) ⊆ 𝑢 ↔ ∀𝑛 ∈ 𝑓 (𝐹‘𝑛) ∈ 𝑢))
75	69, 73, 74	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐿) → ((𝐹 “ 𝑓) ⊆ 𝑢 ↔ ∀𝑛 ∈ 𝑓 (𝐹‘𝑛) ∈ 𝑢))
76	75	rexbidva 3154	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (∃𝑓 ∈ 𝐿 (𝐹 “ 𝑓) ⊆ 𝑢 ↔ ∃𝑓 ∈ 𝐿 ∀𝑛 ∈ 𝑓 (𝐹‘𝑛) ∈ 𝑢))
77		txflf.g	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐺:𝑍⟶𝑌)
78	77	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐿) → 𝐺:𝑍⟶𝑌)
79	78	ffund 6655	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐿) → Fun 𝐺)
80		filelss 23767	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐿 ∈ (Fil‘𝑍) ∧ 𝑔 ∈ 𝐿) → 𝑔 ⊆ 𝑍)
81	23, 80	sylan 580	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐿) → 𝑔 ⊆ 𝑍)
82	78	fdmd 6661	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐿) → dom 𝐺 = 𝑍)
83	81, 82	sseqtrrd 3967	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐿) → 𝑔 ⊆ dom 𝐺)
84		funimass4 6886	. . . . . . . . . . . . . . . . . 18 ⊢ ((Fun 𝐺 ∧ 𝑔 ⊆ dom 𝐺) → ((𝐺 “ 𝑔) ⊆ 𝑣 ↔ ∀𝑛 ∈ 𝑔 (𝐺‘𝑛) ∈ 𝑣))
85	79, 83, 84	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐿) → ((𝐺 “ 𝑔) ⊆ 𝑣 ↔ ∀𝑛 ∈ 𝑔 (𝐺‘𝑛) ∈ 𝑣))
86	85	rexbidva 3154	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (∃𝑔 ∈ 𝐿 (𝐺 “ 𝑔) ⊆ 𝑣 ↔ ∃𝑔 ∈ 𝐿 ∀𝑛 ∈ 𝑔 (𝐺‘𝑛) ∈ 𝑣))
87	76, 86	anbi12d 632	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((∃𝑓 ∈ 𝐿 (𝐹 “ 𝑓) ⊆ 𝑢 ∧ ∃𝑔 ∈ 𝐿 (𝐺 “ 𝑔) ⊆ 𝑣) ↔ (∃𝑓 ∈ 𝐿 ∀𝑛 ∈ 𝑓 (𝐹‘𝑛) ∈ 𝑢 ∧ ∃𝑔 ∈ 𝐿 ∀𝑛 ∈ 𝑔 (𝐺‘𝑛) ∈ 𝑣)))
88	42, 66, 87	3bitr4d 311	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (∃ℎ ∈ 𝐿 (𝐻 “ ℎ) ⊆ (𝑢 × 𝑣) ↔ (∃𝑓 ∈ 𝐿 (𝐹 “ 𝑓) ⊆ 𝑢 ∧ ∃𝑔 ∈ 𝐿 (𝐺 “ 𝑔) ⊆ 𝑣)))
89	14, 88	imbi12d 344	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) → ∃ℎ ∈ 𝐿 (𝐻 “ ℎ) ⊆ (𝑢 × 𝑣)) ↔ ((𝑆 ∈ 𝑣 ∧ 𝑅 ∈ 𝑢) → (∃𝑓 ∈ 𝐿 (𝐹 “ 𝑓) ⊆ 𝑢 ∧ ∃𝑔 ∈ 𝐿 (𝐺 “ 𝑔) ⊆ 𝑣))))
90		impexp 450	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ 𝑣 ∧ 𝑅 ∈ 𝑢) → (∃𝑓 ∈ 𝐿 (𝐹 “ 𝑓) ⊆ 𝑢 ∧ ∃𝑔 ∈ 𝐿 (𝐺 “ 𝑔) ⊆ 𝑣)) ↔ (𝑆 ∈ 𝑣 → (𝑅 ∈ 𝑢 → (∃𝑓 ∈ 𝐿 (𝐹 “ 𝑓) ⊆ 𝑢 ∧ ∃𝑔 ∈ 𝐿 (𝐺 “ 𝑔) ⊆ 𝑣))))
91	89, 90	bitrdi 287	. . . . . . . . . . . 12 ⊢ (𝜑 → ((⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) → ∃ℎ ∈ 𝐿 (𝐻 “ ℎ) ⊆ (𝑢 × 𝑣)) ↔ (𝑆 ∈ 𝑣 → (𝑅 ∈ 𝑢 → (∃𝑓 ∈ 𝐿 (𝐹 “ 𝑓) ⊆ 𝑢 ∧ ∃𝑔 ∈ 𝐿 (𝐺 “ 𝑔) ⊆ 𝑣)))))
92	91	ralbidv 3155	. . . . . . . . . . 11 ⊢ (𝜑 → (∀𝑣 ∈ 𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) → ∃ℎ ∈ 𝐿 (𝐻 “ ℎ) ⊆ (𝑢 × 𝑣)) ↔ ∀𝑣 ∈ 𝐾 (𝑆 ∈ 𝑣 → (𝑅 ∈ 𝑢 → (∃𝑓 ∈ 𝐿 (𝐹 “ 𝑓) ⊆ 𝑢 ∧ ∃𝑔 ∈ 𝐿 (𝐺 “ 𝑔) ⊆ 𝑣)))))
93		eleq2 2820	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑣 → (𝑆 ∈ 𝑥 ↔ 𝑆 ∈ 𝑣))
94	93	ralrab 3648	. . . . . . . . . . . 12 ⊢ (∀𝑣 ∈ {𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥} (𝑅 ∈ 𝑢 → (∃𝑓 ∈ 𝐿 (𝐹 “ 𝑓) ⊆ 𝑢 ∧ ∃𝑔 ∈ 𝐿 (𝐺 “ 𝑔) ⊆ 𝑣)) ↔ ∀𝑣 ∈ 𝐾 (𝑆 ∈ 𝑣 → (𝑅 ∈ 𝑢 → (∃𝑓 ∈ 𝐿 (𝐹 “ 𝑓) ⊆ 𝑢 ∧ ∃𝑔 ∈ 𝐿 (𝐺 “ 𝑔) ⊆ 𝑣))))
95		r19.21v 3157	. . . . . . . . . . . 12 ⊢ (∀𝑣 ∈ {𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥} (𝑅 ∈ 𝑢 → (∃𝑓 ∈ 𝐿 (𝐹 “ 𝑓) ⊆ 𝑢 ∧ ∃𝑔 ∈ 𝐿 (𝐺 “ 𝑔) ⊆ 𝑣)) ↔ (𝑅 ∈ 𝑢 → ∀𝑣 ∈ {𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥} (∃𝑓 ∈ 𝐿 (𝐹 “ 𝑓) ⊆ 𝑢 ∧ ∃𝑔 ∈ 𝐿 (𝐺 “ 𝑔) ⊆ 𝑣)))
96	94, 95	bitr3i 277	. . . . . . . . . . 11 ⊢ (∀𝑣 ∈ 𝐾 (𝑆 ∈ 𝑣 → (𝑅 ∈ 𝑢 → (∃𝑓 ∈ 𝐿 (𝐹 “ 𝑓) ⊆ 𝑢 ∧ ∃𝑔 ∈ 𝐿 (𝐺 “ 𝑔) ⊆ 𝑣))) ↔ (𝑅 ∈ 𝑢 → ∀𝑣 ∈ {𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥} (∃𝑓 ∈ 𝐿 (𝐹 “ 𝑓) ⊆ 𝑢 ∧ ∃𝑔 ∈ 𝐿 (𝐺 “ 𝑔) ⊆ 𝑣)))
97	92, 96	bitrdi 287	. . . . . . . . . 10 ⊢ (𝜑 → (∀𝑣 ∈ 𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) → ∃ℎ ∈ 𝐿 (𝐻 “ ℎ) ⊆ (𝑢 × 𝑣)) ↔ (𝑅 ∈ 𝑢 → ∀𝑣 ∈ {𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥} (∃𝑓 ∈ 𝐿 (𝐹 “ 𝑓) ⊆ 𝑢 ∧ ∃𝑔 ∈ 𝐿 (𝐺 “ 𝑔) ⊆ 𝑣))))
98	97	ralbidv 3155	. . . . . . . . 9 ⊢ (𝜑 → (∀𝑢 ∈ 𝐽 ∀𝑣 ∈ 𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) → ∃ℎ ∈ 𝐿 (𝐻 “ ℎ) ⊆ (𝑢 × 𝑣)) ↔ ∀𝑢 ∈ 𝐽 (𝑅 ∈ 𝑢 → ∀𝑣 ∈ {𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥} (∃𝑓 ∈ 𝐿 (𝐹 “ 𝑓) ⊆ 𝑢 ∧ ∃𝑔 ∈ 𝐿 (𝐺 “ 𝑔) ⊆ 𝑣))))
99		eleq2 2820	. . . . . . . . . 10 ⊢ (𝑥 = 𝑢 → (𝑅 ∈ 𝑥 ↔ 𝑅 ∈ 𝑢))
100	99	ralrab 3648	. . . . . . . . 9 ⊢ (∀𝑢 ∈ {𝑥 ∈ 𝐽 ∣ 𝑅 ∈ 𝑥}∀𝑣 ∈ {𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥} (∃𝑓 ∈ 𝐿 (𝐹 “ 𝑓) ⊆ 𝑢 ∧ ∃𝑔 ∈ 𝐿 (𝐺 “ 𝑔) ⊆ 𝑣) ↔ ∀𝑢 ∈ 𝐽 (𝑅 ∈ 𝑢 → ∀𝑣 ∈ {𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥} (∃𝑓 ∈ 𝐿 (𝐹 “ 𝑓) ⊆ 𝑢 ∧ ∃𝑔 ∈ 𝐿 (𝐺 “ 𝑔) ⊆ 𝑣)))
101	98, 100	bitr4di 289	. . . . . . . 8 ⊢ (𝜑 → (∀𝑢 ∈ 𝐽 ∀𝑣 ∈ 𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) → ∃ℎ ∈ 𝐿 (𝐻 “ ℎ) ⊆ (𝑢 × 𝑣)) ↔ ∀𝑢 ∈ {𝑥 ∈ 𝐽 ∣ 𝑅 ∈ 𝑥}∀𝑣 ∈ {𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥} (∃𝑓 ∈ 𝐿 (𝐹 “ 𝑓) ⊆ 𝑢 ∧ ∃𝑔 ∈ 𝐿 (𝐺 “ 𝑔) ⊆ 𝑣)))
102	101	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝑌)) → (∀𝑢 ∈ 𝐽 ∀𝑣 ∈ 𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) → ∃ℎ ∈ 𝐿 (𝐻 “ ℎ) ⊆ (𝑢 × 𝑣)) ↔ ∀𝑢 ∈ {𝑥 ∈ 𝐽 ∣ 𝑅 ∈ 𝑥}∀𝑣 ∈ {𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥} (∃𝑓 ∈ 𝐿 (𝐹 “ 𝑓) ⊆ 𝑢 ∧ ∃𝑔 ∈ 𝐿 (𝐺 “ 𝑔) ⊆ 𝑣)))
103		txflf.j	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
104		toponmax 22841	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
105	103, 104	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝐽)
106		eleq2 2820	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑋 → (𝑅 ∈ 𝑥 ↔ 𝑅 ∈ 𝑋))
107	106	rspcev 3572	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐽 ∧ 𝑅 ∈ 𝑋) → ∃𝑥 ∈ 𝐽 𝑅 ∈ 𝑥)
108		rabn0 4336	. . . . . . . . . . 11 ⊢ ({𝑥 ∈ 𝐽 ∣ 𝑅 ∈ 𝑥} ≠ ∅ ↔ ∃𝑥 ∈ 𝐽 𝑅 ∈ 𝑥)
109	107, 108	sylibr 234	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝐽 ∧ 𝑅 ∈ 𝑋) → {𝑥 ∈ 𝐽 ∣ 𝑅 ∈ 𝑥} ≠ ∅)
110	105, 109	sylan 580	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑅 ∈ 𝑋) → {𝑥 ∈ 𝐽 ∣ 𝑅 ∈ 𝑥} ≠ ∅)
111		txflf.k	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
112		toponmax 22841	. . . . . . . . . . 11 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 ∈ 𝐾)
113	111, 112	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ 𝐾)
114		eleq2 2820	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑌 → (𝑆 ∈ 𝑥 ↔ 𝑆 ∈ 𝑌))
115	114	rspcev 3572	. . . . . . . . . . 11 ⊢ ((𝑌 ∈ 𝐾 ∧ 𝑆 ∈ 𝑌) → ∃𝑥 ∈ 𝐾 𝑆 ∈ 𝑥)
116		rabn0 4336	. . . . . . . . . . 11 ⊢ ({𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥} ≠ ∅ ↔ ∃𝑥 ∈ 𝐾 𝑆 ∈ 𝑥)
117	115, 116	sylibr 234	. . . . . . . . . 10 ⊢ ((𝑌 ∈ 𝐾 ∧ 𝑆 ∈ 𝑌) → {𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥} ≠ ∅)
118	113, 117	sylan 580	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑆 ∈ 𝑌) → {𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥} ≠ ∅)
119	110, 118	anim12dan 619	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝑌)) → ({𝑥 ∈ 𝐽 ∣ 𝑅 ∈ 𝑥} ≠ ∅ ∧ {𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥} ≠ ∅))
120		r19.28zv 4448	. . . . . . . . . 10 ⊢ ({𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥} ≠ ∅ → (∀𝑣 ∈ {𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥} (∃𝑓 ∈ 𝐿 (𝐹 “ 𝑓) ⊆ 𝑢 ∧ ∃𝑔 ∈ 𝐿 (𝐺 “ 𝑔) ⊆ 𝑣) ↔ (∃𝑓 ∈ 𝐿 (𝐹 “ 𝑓) ⊆ 𝑢 ∧ ∀𝑣 ∈ {𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥}∃𝑔 ∈ 𝐿 (𝐺 “ 𝑔) ⊆ 𝑣)))
121	120	ralbidv 3155	. . . . . . . . 9 ⊢ ({𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥} ≠ ∅ → (∀𝑢 ∈ {𝑥 ∈ 𝐽 ∣ 𝑅 ∈ 𝑥}∀𝑣 ∈ {𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥} (∃𝑓 ∈ 𝐿 (𝐹 “ 𝑓) ⊆ 𝑢 ∧ ∃𝑔 ∈ 𝐿 (𝐺 “ 𝑔) ⊆ 𝑣) ↔ ∀𝑢 ∈ {𝑥 ∈ 𝐽 ∣ 𝑅 ∈ 𝑥} (∃𝑓 ∈ 𝐿 (𝐹 “ 𝑓) ⊆ 𝑢 ∧ ∀𝑣 ∈ {𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥}∃𝑔 ∈ 𝐿 (𝐺 “ 𝑔) ⊆ 𝑣)))
122		r19.27zv 4453	. . . . . . . . 9 ⊢ ({𝑥 ∈ 𝐽 ∣ 𝑅 ∈ 𝑥} ≠ ∅ → (∀𝑢 ∈ {𝑥 ∈ 𝐽 ∣ 𝑅 ∈ 𝑥} (∃𝑓 ∈ 𝐿 (𝐹 “ 𝑓) ⊆ 𝑢 ∧ ∀𝑣 ∈ {𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥}∃𝑔 ∈ 𝐿 (𝐺 “ 𝑔) ⊆ 𝑣) ↔ (∀𝑢 ∈ {𝑥 ∈ 𝐽 ∣ 𝑅 ∈ 𝑥}∃𝑓 ∈ 𝐿 (𝐹 “ 𝑓) ⊆ 𝑢 ∧ ∀𝑣 ∈ {𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥}∃𝑔 ∈ 𝐿 (𝐺 “ 𝑔) ⊆ 𝑣)))
123	121, 122	sylan9bbr 510	. . . . . . . 8 ⊢ (({𝑥 ∈ 𝐽 ∣ 𝑅 ∈ 𝑥} ≠ ∅ ∧ {𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥} ≠ ∅) → (∀𝑢 ∈ {𝑥 ∈ 𝐽 ∣ 𝑅 ∈ 𝑥}∀𝑣 ∈ {𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥} (∃𝑓 ∈ 𝐿 (𝐹 “ 𝑓) ⊆ 𝑢 ∧ ∃𝑔 ∈ 𝐿 (𝐺 “ 𝑔) ⊆ 𝑣) ↔ (∀𝑢 ∈ {𝑥 ∈ 𝐽 ∣ 𝑅 ∈ 𝑥}∃𝑓 ∈ 𝐿 (𝐹 “ 𝑓) ⊆ 𝑢 ∧ ∀𝑣 ∈ {𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥}∃𝑔 ∈ 𝐿 (𝐺 “ 𝑔) ⊆ 𝑣)))
124	119, 123	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝑌)) → (∀𝑢 ∈ {𝑥 ∈ 𝐽 ∣ 𝑅 ∈ 𝑥}∀𝑣 ∈ {𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥} (∃𝑓 ∈ 𝐿 (𝐹 “ 𝑓) ⊆ 𝑢 ∧ ∃𝑔 ∈ 𝐿 (𝐺 “ 𝑔) ⊆ 𝑣) ↔ (∀𝑢 ∈ {𝑥 ∈ 𝐽 ∣ 𝑅 ∈ 𝑥}∃𝑓 ∈ 𝐿 (𝐹 “ 𝑓) ⊆ 𝑢 ∧ ∀𝑣 ∈ {𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥}∃𝑔 ∈ 𝐿 (𝐺 “ 𝑔) ⊆ 𝑣)))
125	102, 124	bitrd 279	. . . . . 6 ⊢ ((𝜑 ∧ (𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝑌)) → (∀𝑢 ∈ 𝐽 ∀𝑣 ∈ 𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) → ∃ℎ ∈ 𝐿 (𝐻 “ ℎ) ⊆ (𝑢 × 𝑣)) ↔ (∀𝑢 ∈ {𝑥 ∈ 𝐽 ∣ 𝑅 ∈ 𝑥}∃𝑓 ∈ 𝐿 (𝐹 “ 𝑓) ⊆ 𝑢 ∧ ∀𝑣 ∈ {𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥}∃𝑔 ∈ 𝐿 (𝐺 “ 𝑔) ⊆ 𝑣)))
126	99	ralrab 3648	. . . . . . 7 ⊢ (∀𝑢 ∈ {𝑥 ∈ 𝐽 ∣ 𝑅 ∈ 𝑥}∃𝑓 ∈ 𝐿 (𝐹 “ 𝑓) ⊆ 𝑢 ↔ ∀𝑢 ∈ 𝐽 (𝑅 ∈ 𝑢 → ∃𝑓 ∈ 𝐿 (𝐹 “ 𝑓) ⊆ 𝑢))
127	93	ralrab 3648	. . . . . . 7 ⊢ (∀𝑣 ∈ {𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥}∃𝑔 ∈ 𝐿 (𝐺 “ 𝑔) ⊆ 𝑣 ↔ ∀𝑣 ∈ 𝐾 (𝑆 ∈ 𝑣 → ∃𝑔 ∈ 𝐿 (𝐺 “ 𝑔) ⊆ 𝑣))
128	126, 127	anbi12i 628	. . . . . 6 ⊢ ((∀𝑢 ∈ {𝑥 ∈ 𝐽 ∣ 𝑅 ∈ 𝑥}∃𝑓 ∈ 𝐿 (𝐹 “ 𝑓) ⊆ 𝑢 ∧ ∀𝑣 ∈ {𝑥 ∈ 𝐾 ∣ 𝑆 ∈ 𝑥}∃𝑔 ∈ 𝐿 (𝐺 “ 𝑔) ⊆ 𝑣) ↔ (∀𝑢 ∈ 𝐽 (𝑅 ∈ 𝑢 → ∃𝑓 ∈ 𝐿 (𝐹 “ 𝑓) ⊆ 𝑢) ∧ ∀𝑣 ∈ 𝐾 (𝑆 ∈ 𝑣 → ∃𝑔 ∈ 𝐿 (𝐺 “ 𝑔) ⊆ 𝑣)))
129	125, 128	bitrdi 287	. . . . 5 ⊢ ((𝜑 ∧ (𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝑌)) → (∀𝑢 ∈ 𝐽 ∀𝑣 ∈ 𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) → ∃ℎ ∈ 𝐿 (𝐻 “ ℎ) ⊆ (𝑢 × 𝑣)) ↔ (∀𝑢 ∈ 𝐽 (𝑅 ∈ 𝑢 → ∃𝑓 ∈ 𝐿 (𝐹 “ 𝑓) ⊆ 𝑢) ∧ ∀𝑣 ∈ 𝐾 (𝑆 ∈ 𝑣 → ∃𝑔 ∈ 𝐿 (𝐺 “ 𝑔) ⊆ 𝑣))))
130	11, 129	bitrid 283	. . . 4 ⊢ ((𝜑 ∧ (𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝑌)) → (∀𝑧 ∈ ran (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣))(⟨𝑅, 𝑆⟩ ∈ 𝑧 → ∃ℎ ∈ 𝐿 (𝐻 “ ℎ) ⊆ 𝑧) ↔ (∀𝑢 ∈ 𝐽 (𝑅 ∈ 𝑢 → ∃𝑓 ∈ 𝐿 (𝐹 “ 𝑓) ⊆ 𝑢) ∧ ∀𝑣 ∈ 𝐾 (𝑆 ∈ 𝑣 → ∃𝑔 ∈ 𝐿 (𝐺 “ 𝑔) ⊆ 𝑣))))
131	130	pm5.32da 579	. . 3 ⊢ (𝜑 → (((𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝑌) ∧ ∀𝑧 ∈ ran (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣))(⟨𝑅, 𝑆⟩ ∈ 𝑧 → ∃ℎ ∈ 𝐿 (𝐻 “ ℎ) ⊆ 𝑧)) ↔ ((𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝑌) ∧ (∀𝑢 ∈ 𝐽 (𝑅 ∈ 𝑢 → ∃𝑓 ∈ 𝐿 (𝐹 “ 𝑓) ⊆ 𝑢) ∧ ∀𝑣 ∈ 𝐾 (𝑆 ∈ 𝑣 → ∃𝑔 ∈ 𝐿 (𝐺 “ 𝑔) ⊆ 𝑣)))))
132		opelxp 5650	. . . 4 ⊢ (⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑌) ↔ (𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝑌))
133	132	anbi1i 624	. . 3 ⊢ ((⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑌) ∧ ∀𝑧 ∈ ran (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣))(⟨𝑅, 𝑆⟩ ∈ 𝑧 → ∃ℎ ∈ 𝐿 (𝐻 “ ℎ) ⊆ 𝑧)) ↔ ((𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝑌) ∧ ∀𝑧 ∈ ran (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣))(⟨𝑅, 𝑆⟩ ∈ 𝑧 → ∃ℎ ∈ 𝐿 (𝐻 “ ℎ) ⊆ 𝑧)))
134		an4 656	. . 3 ⊢ (((𝑅 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑅 ∈ 𝑢 → ∃𝑓 ∈ 𝐿 (𝐹 “ 𝑓) ⊆ 𝑢)) ∧ (𝑆 ∈ 𝑌 ∧ ∀𝑣 ∈ 𝐾 (𝑆 ∈ 𝑣 → ∃𝑔 ∈ 𝐿 (𝐺 “ 𝑔) ⊆ 𝑣))) ↔ ((𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝑌) ∧ (∀𝑢 ∈ 𝐽 (𝑅 ∈ 𝑢 → ∃𝑓 ∈ 𝐿 (𝐹 “ 𝑓) ⊆ 𝑢) ∧ ∀𝑣 ∈ 𝐾 (𝑆 ∈ 𝑣 → ∃𝑔 ∈ 𝐿 (𝐺 “ 𝑔) ⊆ 𝑣))))
135	131, 133, 134	3bitr4g 314	. 2 ⊢ (𝜑 → ((⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑌) ∧ ∀𝑧 ∈ ran (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣))(⟨𝑅, 𝑆⟩ ∈ 𝑧 → ∃ℎ ∈ 𝐿 (𝐻 “ ℎ) ⊆ 𝑧)) ↔ ((𝑅 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑅 ∈ 𝑢 → ∃𝑓 ∈ 𝐿 (𝐹 “ 𝑓) ⊆ 𝑢)) ∧ (𝑆 ∈ 𝑌 ∧ ∀𝑣 ∈ 𝐾 (𝑆 ∈ 𝑣 → ∃𝑔 ∈ 𝐿 (𝐺 “ 𝑔) ⊆ 𝑣)))))
136		eqid 2731	. . . . . . . 8 ⊢ ran (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣)) = ran (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣))
137	136	txval 23479	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×_t 𝐾) = (topGen‘ran (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣))))
138	103, 111, 137	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝐽 ×_t 𝐾) = (topGen‘ran (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣))))
139	138	oveq1d 7361	. . . . 5 ⊢ (𝜑 → ((𝐽 ×_t 𝐾) fLimf 𝐿) = ((topGen‘ran (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣))) fLimf 𝐿))
140	139	fveq1d 6824	. . . 4 ⊢ (𝜑 → (((𝐽 ×_t 𝐾) fLimf 𝐿)‘𝐻) = (((topGen‘ran (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣))) fLimf 𝐿)‘𝐻))
141	140	eleq2d 2817	. . 3 ⊢ (𝜑 → (⟨𝑅, 𝑆⟩ ∈ (((𝐽 ×_t 𝐾) fLimf 𝐿)‘𝐻) ↔ ⟨𝑅, 𝑆⟩ ∈ (((topGen‘ran (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣))) fLimf 𝐿)‘𝐻)))
142		txtopon 23506	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×_t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
143	103, 111, 142	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝐽 ×_t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
144	138, 143	eqeltrrd 2832	. . . 4 ⊢ (𝜑 → (topGen‘ran (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣))) ∈ (TopOn‘(𝑋 × 𝑌)))
145	67	ffvelcdmda 7017	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ 𝑋)
146	77	ffvelcdmda 7017	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐺‘𝑛) ∈ 𝑌)
147	145, 146	opelxpd 5653	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ⟨(𝐹‘𝑛), (𝐺‘𝑛)⟩ ∈ (𝑋 × 𝑌))
148	147, 46	fmptd 7047	. . . 4 ⊢ (𝜑 → 𝐻:𝑍⟶(𝑋 × 𝑌))
149		eqid 2731	. . . . 5 ⊢ (topGen‘ran (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣))) = (topGen‘ran (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣)))
150	149	flftg 23911	. . . 4 ⊢ (((topGen‘ran (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣))) ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝐿 ∈ (Fil‘𝑍) ∧ 𝐻:𝑍⟶(𝑋 × 𝑌)) → (⟨𝑅, 𝑆⟩ ∈ (((topGen‘ran (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣))) fLimf 𝐿)‘𝐻) ↔ (⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑌) ∧ ∀𝑧 ∈ ran (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣))(⟨𝑅, 𝑆⟩ ∈ 𝑧 → ∃ℎ ∈ 𝐿 (𝐻 “ ℎ) ⊆ 𝑧))))
151	144, 23, 148, 150	syl3anc 1373	. . 3 ⊢ (𝜑 → (⟨𝑅, 𝑆⟩ ∈ (((topGen‘ran (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣))) fLimf 𝐿)‘𝐻) ↔ (⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑌) ∧ ∀𝑧 ∈ ran (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣))(⟨𝑅, 𝑆⟩ ∈ 𝑧 → ∃ℎ ∈ 𝐿 (𝐻 “ ℎ) ⊆ 𝑧))))
152	141, 151	bitrd 279	. 2 ⊢ (𝜑 → (⟨𝑅, 𝑆⟩ ∈ (((𝐽 ×_t 𝐾) fLimf 𝐿)‘𝐻) ↔ (⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑌) ∧ ∀𝑧 ∈ ran (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣))(⟨𝑅, 𝑆⟩ ∈ 𝑧 → ∃ℎ ∈ 𝐿 (𝐻 “ ℎ) ⊆ 𝑧))))
153		isflf 23908	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑍) ∧ 𝐹:𝑍⟶𝑋) → (𝑅 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ (𝑅 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑅 ∈ 𝑢 → ∃𝑓 ∈ 𝐿 (𝐹 “ 𝑓) ⊆ 𝑢))))
154	103, 23, 67, 153	syl3anc 1373	. . 3 ⊢ (𝜑 → (𝑅 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ (𝑅 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑅 ∈ 𝑢 → ∃𝑓 ∈ 𝐿 (𝐹 “ 𝑓) ⊆ 𝑢))))
155		isflf 23908	. . . 4 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (Fil‘𝑍) ∧ 𝐺:𝑍⟶𝑌) → (𝑆 ∈ ((𝐾 fLimf 𝐿)‘𝐺) ↔ (𝑆 ∈ 𝑌 ∧ ∀𝑣 ∈ 𝐾 (𝑆 ∈ 𝑣 → ∃𝑔 ∈ 𝐿 (𝐺 “ 𝑔) ⊆ 𝑣))))
156	111, 23, 77, 155	syl3anc 1373	. . 3 ⊢ (𝜑 → (𝑆 ∈ ((𝐾 fLimf 𝐿)‘𝐺) ↔ (𝑆 ∈ 𝑌 ∧ ∀𝑣 ∈ 𝐾 (𝑆 ∈ 𝑣 → ∃𝑔 ∈ 𝐿 (𝐺 “ 𝑔) ⊆ 𝑣))))
157	154, 156	anbi12d 632	. 2 ⊢ (𝜑 → ((𝑅 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝑆 ∈ ((𝐾 fLimf 𝐿)‘𝐺)) ↔ ((𝑅 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑅 ∈ 𝑢 → ∃𝑓 ∈ 𝐿 (𝐹 “ 𝑓) ⊆ 𝑢)) ∧ (𝑆 ∈ 𝑌 ∧ ∀𝑣 ∈ 𝐾 (𝑆 ∈ 𝑣 → ∃𝑔 ∈ 𝐿 (𝐺 “ 𝑔) ⊆ 𝑣)))))
158	135, 152, 157	3bitr4d 311	1 ⊢ (𝜑 → (⟨𝑅, 𝑆⟩ ∈ (((𝐽 ×_t 𝐾) fLimf 𝐿)‘𝐻) ↔ (𝑅 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝑆 ∈ ((𝐾 fLimf 𝐿)‘𝐺))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 ∀wral 3047 ∃wrex 3056 {crab 3395 Vcvv 3436 ∩ cin 3896 ⊆ wss 3897 ∅c0 4280 ⟨cop 4579 ↦ cmpt 5170 × cxp 5612 dom cdm 5614 ran crn 5615 ↾ cres 5616 “ cima 5617 Fun wfun 6475 Fn wfn 6476 ⟶wf 6477 ‘cfv 6481 (class class class)co 7346 ∈ cmpo 7348 topGenctg 17341 TopOnctopon 22825 ×_t ctx 23475 Filcfil 23760 fLimf cflf 23850

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 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-id 5509 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-ov 7349 df-oprab 7350 df-mpo 7351 df-1st 7921 df-2nd 7922 df-map 8752 df-topgen 17347 df-fbas 21288 df-fg 21289 df-top 22809 df-topon 22826 df-bases 22861 df-ntr 22935 df-nei 23013 df-tx 23477 df-fil 23761 df-fm 23853 df-flim 23854 df-flf 23855

This theorem is referenced by: flfcnp2 23922

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