Theorem txlm 21451

Description: Two sequences converge iff the sequence of their ordered pairs converges. Proposition 14-2.6 of [Gleason] p. 230. (Contributed by NM, 16-Jul-2007.) (Revised by Mario Carneiro, 5-May-2014.)

Hypotheses

Ref	Expression
txlm.z	⊢ 𝑍 = (ℤ≥‘𝑀)
txlm.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
txlm.j	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
txlm.k	⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
txlm.f	⊢ (𝜑 → 𝐹:𝑍⟶𝑋)
txlm.g	⊢ (𝜑 → 𝐺:𝑍⟶𝑌)
txlm.h	⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ ⟨(𝐹‘𝑛), (𝐺‘𝑛)⟩)

Assertion

Ref	Expression
txlm	⊢ (𝜑 → ((𝐹(⇝𝑡‘𝐽)𝑅 ∧ 𝐺(⇝𝑡‘𝐾)𝑆) ↔ 𝐻(⇝𝑡‘(𝐽 ×t 𝐾))⟨𝑅, 𝑆⟩))

Distinct variable groups:   𝑛,𝐹   𝜑,𝑛   𝑛,𝐺   𝑛,𝐽   𝑛,𝐾   𝑛,𝑋   𝑛,𝑌   𝑛,𝑍

Allowed substitution hints:   𝑅(𝑛)   𝑆(𝑛)   𝐻(𝑛)   𝑀(𝑛)

Proof of Theorem txlm

Dummy variables 𝑗 𝑘 𝑢 𝑣 𝑤 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		r19.27v 3070	. . . . . . . 8 ⊢ ((∀𝑢 ∈ 𝐽 (𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ ∀𝑣 ∈ 𝐾 (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)) → ∀𝑢 ∈ 𝐽 ((𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ ∀𝑣 ∈ 𝐾 (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)))
2		r19.28v 3071	. . . . . . . . 9 ⊢ (((𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ ∀𝑣 ∈ 𝐾 (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)) → ∀𝑣 ∈ 𝐾 ((𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)))
3	2	ralimi 2952	. . . . . . . 8 ⊢ (∀𝑢 ∈ 𝐽 ((𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ ∀𝑣 ∈ 𝐾 (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)) → ∀𝑢 ∈ 𝐽 ∀𝑣 ∈ 𝐾 ((𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)))
4	1, 3	syl 17	. . . . . . 7 ⊢ ((∀𝑢 ∈ 𝐽 (𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ ∀𝑣 ∈ 𝐾 (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)) → ∀𝑢 ∈ 𝐽 ∀𝑣 ∈ 𝐾 ((𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)))
5		simprl 794	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑤 ∈ (𝐽 ×t 𝐾) ∧ ⟨𝑅, 𝑆⟩ ∈ 𝑤)) → 𝑤 ∈ (𝐽 ×t 𝐾))
6		txlm.j	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
7		topontop 20718	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
8	6, 7	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐽 ∈ Top)
9		txlm.k	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
10		topontop 20718	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
11	9, 10	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐾 ∈ Top)
12		eqid 2622	. . . . . . . . . . . . . . . . . 18 ⊢ ran (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣)) = ran (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣))
13	12	txval 21367	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ×t 𝐾) = (topGen‘ran (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣))))
14	8, 11, 13	syl2anc 693	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐽 ×t 𝐾) = (topGen‘ran (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣))))
15	14	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑤 ∈ (𝐽 ×t 𝐾) ∧ ⟨𝑅, 𝑆⟩ ∈ 𝑤)) → (𝐽 ×t 𝐾) = (topGen‘ran (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣))))
16	5, 15	eleqtrd 2703	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑤 ∈ (𝐽 ×t 𝐾) ∧ ⟨𝑅, 𝑆⟩ ∈ 𝑤)) → 𝑤 ∈ (topGen‘ran (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣))))
17		simprr 796	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑤 ∈ (𝐽 ×t 𝐾) ∧ ⟨𝑅, 𝑆⟩ ∈ 𝑤)) → ⟨𝑅, 𝑆⟩ ∈ 𝑤)
18		tg2 20769	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ (topGen‘ran (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣))) ∧ ⟨𝑅, 𝑆⟩ ∈ 𝑤) → ∃𝑡 ∈ ran (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣))(⟨𝑅, 𝑆⟩ ∈ 𝑡 ∧ 𝑡 ⊆ 𝑤))
19	16, 17, 18	syl2anc 693	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑤 ∈ (𝐽 ×t 𝐾) ∧ ⟨𝑅, 𝑆⟩ ∈ 𝑤)) → ∃𝑡 ∈ ran (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣))(⟨𝑅, 𝑆⟩ ∈ 𝑡 ∧ 𝑡 ⊆ 𝑤))
20		vex 3203	. . . . . . . . . . . . . . . 16 ⊢ 𝑢 ∈ V
21		vex 3203	. . . . . . . . . . . . . . . 16 ⊢ 𝑣 ∈ V
22	20, 21	xpex 6962	. . . . . . . . . . . . . . 15 ⊢ (𝑢 × 𝑣) ∈ V
23	22	rgen2w 2925	. . . . . . . . . . . . . 14 ⊢ ∀𝑢 ∈ 𝐽 ∀𝑣 ∈ 𝐾 (𝑢 × 𝑣) ∈ V
24		eqid 2622	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣)) = (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣))
25		eleq2 2690	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = (𝑢 × 𝑣) → (⟨𝑅, 𝑆⟩ ∈ 𝑡 ↔ ⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣)))
26		sseq1 3626	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = (𝑢 × 𝑣) → (𝑡 ⊆ 𝑤 ↔ (𝑢 × 𝑣) ⊆ 𝑤))
27	25, 26	anbi12d 747	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = (𝑢 × 𝑣) → ((⟨𝑅, 𝑆⟩ ∈ 𝑡 ∧ 𝑡 ⊆ 𝑤) ↔ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)))
28	24, 27	rexrnmpt2 6776	. . . . . . . . . . . . . 14 ⊢ (∀𝑢 ∈ 𝐽 ∀𝑣 ∈ 𝐾 (𝑢 × 𝑣) ∈ V → (∃𝑡 ∈ ran (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣))(⟨𝑅, 𝑆⟩ ∈ 𝑡 ∧ 𝑡 ⊆ 𝑤) ↔ ∃𝑢 ∈ 𝐽 ∃𝑣 ∈ 𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)))
29	23, 28	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (∃𝑡 ∈ ran (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣))(⟨𝑅, 𝑆⟩ ∈ 𝑡 ∧ 𝑡 ⊆ 𝑤) ↔ ∃𝑢 ∈ 𝐽 ∃𝑣 ∈ 𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤))
30	19, 29	sylib 208	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑤 ∈ (𝐽 ×t 𝐾) ∧ ⟨𝑅, 𝑆⟩ ∈ 𝑤)) → ∃𝑢 ∈ 𝐽 ∃𝑣 ∈ 𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤))
31	30	ex 450	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑤 ∈ (𝐽 ×t 𝐾) ∧ ⟨𝑅, 𝑆⟩ ∈ 𝑤) → ∃𝑢 ∈ 𝐽 ∃𝑣 ∈ 𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)))
32		r19.29 3072	. . . . . . . . . . . . 13 ⊢ ((∀𝑢 ∈ 𝐽 ∀𝑣 ∈ 𝐾 ((𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)) ∧ ∃𝑢 ∈ 𝐽 ∃𝑣 ∈ 𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) → ∃𝑢 ∈ 𝐽 (∀𝑣 ∈ 𝐾 ((𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)) ∧ ∃𝑣 ∈ 𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)))
33		r19.29 3072	. . . . . . . . . . . . . . 15 ⊢ ((∀𝑣 ∈ 𝐾 ((𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)) ∧ ∃𝑣 ∈ 𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) → ∃𝑣 ∈ 𝐾 (((𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)))
34		simprl 794	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑣 ∈ 𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) → ⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣))
35		opelxp 5146	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ↔ (𝑅 ∈ 𝑢 ∧ 𝑆 ∈ 𝑣))
36	34, 35	sylib 208	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑣 ∈ 𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) → (𝑅 ∈ 𝑢 ∧ 𝑆 ∈ 𝑣))
37		pm2.27 42	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑅 ∈ 𝑢 → ((𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢))
38		pm2.27 42	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑆 ∈ 𝑣 → ((𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣))
39	37, 38	im2anan9 880	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅 ∈ 𝑢 ∧ 𝑆 ∈ 𝑣) → (((𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)) → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢 ∧ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)))
40	36, 39	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑣 ∈ 𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) → (((𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)) → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢 ∧ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)))
41		txlm.z	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑍 = (ℤ≥‘𝑀)
42	41	rexanuz2 14089	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ 𝑢 ∧ (𝐺‘𝑘) ∈ 𝑣) ↔ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢 ∧ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣))
43	41	uztrn2 11705	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → 𝑘 ∈ 𝑍)
44		opelxpi 5148	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝐹‘𝑘) ∈ 𝑢 ∧ (𝐺‘𝑘) ∈ 𝑣) → ⟨(𝐹‘𝑘), (𝐺‘𝑘)⟩ ∈ (𝑢 × 𝑣))
45		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑛 = 𝑘 → (𝐹‘𝑛) = (𝐹‘𝑘))
46		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑛 = 𝑘 → (𝐺‘𝑛) = (𝐺‘𝑘))
47	45, 46	opeq12d 4410	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑛 = 𝑘 → ⟨(𝐹‘𝑛), (𝐺‘𝑛)⟩ = ⟨(𝐹‘𝑘), (𝐺‘𝑘)⟩)
48		txlm.h	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ ⟨(𝐹‘𝑛), (𝐺‘𝑛)⟩)
49		opex 4932	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ⟨(𝐹‘𝑘), (𝐺‘𝑘)⟩ ∈ V
50	47, 48, 49	fvmpt 6282	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑘 ∈ 𝑍 → (𝐻‘𝑘) = ⟨(𝐹‘𝑘), (𝐺‘𝑘)⟩)
51	50	eleq1d 2686	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑘 ∈ 𝑍 → ((𝐻‘𝑘) ∈ (𝑢 × 𝑣) ↔ ⟨(𝐹‘𝑘), (𝐺‘𝑘)⟩ ∈ (𝑢 × 𝑣)))
52	44, 51	syl5ibr 236	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑘 ∈ 𝑍 → (((𝐹‘𝑘) ∈ 𝑢 ∧ (𝐺‘𝑘) ∈ 𝑣) → (𝐻‘𝑘) ∈ (𝑢 × 𝑣)))
53	52	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑣 ∈ 𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) ∧ 𝑘 ∈ 𝑍) → (((𝐹‘𝑘) ∈ 𝑢 ∧ (𝐺‘𝑘) ∈ 𝑣) → (𝐻‘𝑘) ∈ (𝑢 × 𝑣)))
54		simplrr 801	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑣 ∈ 𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) ∧ 𝑘 ∈ 𝑍) → (𝑢 × 𝑣) ⊆ 𝑤)
55	54	sseld 3602	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑣 ∈ 𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) ∧ 𝑘 ∈ 𝑍) → ((𝐻‘𝑘) ∈ (𝑢 × 𝑣) → (𝐻‘𝑘) ∈ 𝑤))
56	53, 55	syld 47	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑣 ∈ 𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) ∧ 𝑘 ∈ 𝑍) → (((𝐹‘𝑘) ∈ 𝑢 ∧ (𝐺‘𝑘) ∈ 𝑣) → (𝐻‘𝑘) ∈ 𝑤))
57	43, 56	sylan2 491	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑣 ∈ 𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘𝑗))) → (((𝐹‘𝑘) ∈ 𝑢 ∧ (𝐺‘𝑘) ∈ 𝑣) → (𝐻‘𝑘) ∈ 𝑤))
58	57	anassrs 680	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑣 ∈ 𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (((𝐹‘𝑘) ∈ 𝑢 ∧ (𝐺‘𝑘) ∈ 𝑣) → (𝐻‘𝑘) ∈ 𝑤))
59	58	ralimdva 2962	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑣 ∈ 𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) ∧ 𝑗 ∈ 𝑍) → (∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ 𝑢 ∧ (𝐺‘𝑘) ∈ 𝑣) → ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤))
60	59	reximdva 3017	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑣 ∈ 𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ 𝑢 ∧ (𝐺‘𝑘) ∈ 𝑣) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤))
61	42, 60	syl5bir 233	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑣 ∈ 𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) → ((∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢 ∧ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤))
62	40, 61	syld 47	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑣 ∈ 𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) → (((𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤))
63	62	ex 450	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑣 ∈ 𝐾) → ((⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) → (((𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤)))
64	63	com23 86	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑣 ∈ 𝐾) → (((𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)) → ((⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤)))
65	64	impd 447	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑣 ∈ 𝐾) → ((((𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤))
66	65	rexlimdva 3031	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐽) → (∃𝑣 ∈ 𝐾 (((𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤))
67	33, 66	syl5 34	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐽) → ((∀𝑣 ∈ 𝐾 ((𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)) ∧ ∃𝑣 ∈ 𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤))
68	67	rexlimdva 3031	. . . . . . . . . . . . 13 ⊢ (𝜑 → (∃𝑢 ∈ 𝐽 (∀𝑣 ∈ 𝐾 ((𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)) ∧ ∃𝑣 ∈ 𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤))
69	32, 68	syl5 34	. . . . . . . . . . . 12 ⊢ (𝜑 → ((∀𝑢 ∈ 𝐽 ∀𝑣 ∈ 𝐾 ((𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)) ∧ ∃𝑢 ∈ 𝐽 ∃𝑣 ∈ 𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤))
70	69	expcomd 454	. . . . . . . . . . 11 ⊢ (𝜑 → (∃𝑢 ∈ 𝐽 ∃𝑣 ∈ 𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) → (∀𝑢 ∈ 𝐽 ∀𝑣 ∈ 𝐾 ((𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤)))
71	31, 70	syld 47	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑤 ∈ (𝐽 ×t 𝐾) ∧ ⟨𝑅, 𝑆⟩ ∈ 𝑤) → (∀𝑢 ∈ 𝐽 ∀𝑣 ∈ 𝐾 ((𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤)))
72	71	expdimp 453	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐽 ×t 𝐾)) → (⟨𝑅, 𝑆⟩ ∈ 𝑤 → (∀𝑢 ∈ 𝐽 ∀𝑣 ∈ 𝐾 ((𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤)))
73	72	com23 86	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐽 ×t 𝐾)) → (∀𝑢 ∈ 𝐽 ∀𝑣 ∈ 𝐾 ((𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)) → (⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤)))
74	73	ralrimdva 2969	. . . . . . 7 ⊢ (𝜑 → (∀𝑢 ∈ 𝐽 ∀𝑣 ∈ 𝐾 ((𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)) → ∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤)))
75	4, 74	syl5 34	. . . . . 6 ⊢ (𝜑 → ((∀𝑢 ∈ 𝐽 (𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ ∀𝑣 ∈ 𝐾 (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)) → ∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤)))
76	75	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝑌)) → ((∀𝑢 ∈ 𝐽 (𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ ∀𝑣 ∈ 𝐾 (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)) → ∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤)))
77	8	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑆 ∈ 𝑌 ∧ 𝑢 ∈ 𝐽)) → 𝐽 ∈ Top)
78	11	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑆 ∈ 𝑌 ∧ 𝑢 ∈ 𝐽)) → 𝐾 ∈ Top)
79		simprr 796	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑆 ∈ 𝑌 ∧ 𝑢 ∈ 𝐽)) → 𝑢 ∈ 𝐽)
80		toponmax 20730	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 ∈ 𝐾)
81	9, 80	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 ∈ 𝐾)
82	81	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑆 ∈ 𝑌 ∧ 𝑢 ∈ 𝐽)) → 𝑌 ∈ 𝐾)
83		txopn 21405	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑢 ∈ 𝐽 ∧ 𝑌 ∈ 𝐾)) → (𝑢 × 𝑌) ∈ (𝐽 ×t 𝐾))
84	77, 78, 79, 82, 83	syl22anc 1327	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑆 ∈ 𝑌 ∧ 𝑢 ∈ 𝐽)) → (𝑢 × 𝑌) ∈ (𝐽 ×t 𝐾))
85		eleq2 2690	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝑢 × 𝑌) → (⟨𝑅, 𝑆⟩ ∈ 𝑤 ↔ ⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑌)))
86		eleq2 2690	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (𝑢 × 𝑌) → ((𝐻‘𝑘) ∈ 𝑤 ↔ (𝐻‘𝑘) ∈ (𝑢 × 𝑌)))
87	86	rexralbidv 3058	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝑢 × 𝑌) → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤 ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ (𝑢 × 𝑌)))
88	85, 87	imbi12d 334	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝑢 × 𝑌) → ((⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤) ↔ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑌) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ (𝑢 × 𝑌))))
89	88	rspcv 3305	. . . . . . . . . . 11 ⊢ ((𝑢 × 𝑌) ∈ (𝐽 ×t 𝐾) → (∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤) → (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑌) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ (𝑢 × 𝑌))))
90	84, 89	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑆 ∈ 𝑌 ∧ 𝑢 ∈ 𝐽)) → (∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤) → (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑌) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ (𝑢 × 𝑌))))
91		simprl 794	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑆 ∈ 𝑌 ∧ 𝑢 ∈ 𝐽)) → 𝑆 ∈ 𝑌)
92		opelxpi 5148	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ 𝑢 ∧ 𝑆 ∈ 𝑌) → ⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑌))
93	91, 92	sylan2 491	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ 𝑢 ∧ (𝜑 ∧ (𝑆 ∈ 𝑌 ∧ 𝑢 ∈ 𝐽))) → ⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑌))
94	93	expcom 451	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑆 ∈ 𝑌 ∧ 𝑢 ∈ 𝐽)) → (𝑅 ∈ 𝑢 → ⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑌)))
95	50	eleq1d 2686	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ 𝑍 → ((𝐻‘𝑘) ∈ (𝑢 × 𝑌) ↔ ⟨(𝐹‘𝑘), (𝐺‘𝑘)⟩ ∈ (𝑢 × 𝑌)))
96		opelxp1 5150	. . . . . . . . . . . . . . . 16 ⊢ (⟨(𝐹‘𝑘), (𝐺‘𝑘)⟩ ∈ (𝑢 × 𝑌) → (𝐹‘𝑘) ∈ 𝑢)
97	95, 96	syl6bi 243	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ 𝑍 → ((𝐻‘𝑘) ∈ (𝑢 × 𝑌) → (𝐹‘𝑘) ∈ 𝑢))
98	43, 97	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → ((𝐻‘𝑘) ∈ (𝑢 × 𝑌) → (𝐹‘𝑘) ∈ 𝑢))
99	98	ralimdva 2962	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ 𝑍 → (∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ (𝑢 × 𝑌) → ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢))
100	99	reximia 3009	. . . . . . . . . . . 12 ⊢ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ (𝑢 × 𝑌) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢)
101	100	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑆 ∈ 𝑌 ∧ 𝑢 ∈ 𝐽)) → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ (𝑢 × 𝑌) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢))
102	94, 101	imim12d 81	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑆 ∈ 𝑌 ∧ 𝑢 ∈ 𝐽)) → ((⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑌) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ (𝑢 × 𝑌)) → (𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢)))
103	90, 102	syld 47	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑆 ∈ 𝑌 ∧ 𝑢 ∈ 𝐽)) → (∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤) → (𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢)))
104	103	anassrs 680	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑆 ∈ 𝑌) ∧ 𝑢 ∈ 𝐽) → (∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤) → (𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢)))
105	104	ralrimdva 2969	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑆 ∈ 𝑌) → (∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤) → ∀𝑢 ∈ 𝐽 (𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢)))
106	105	adantrl 752	. . . . . 6 ⊢ ((𝜑 ∧ (𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝑌)) → (∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤) → ∀𝑢 ∈ 𝐽 (𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢)))
107	8	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑅 ∈ 𝑋 ∧ 𝑣 ∈ 𝐾)) → 𝐽 ∈ Top)
108	11	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑅 ∈ 𝑋 ∧ 𝑣 ∈ 𝐾)) → 𝐾 ∈ Top)
109		toponmax 20730	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
110	6, 109	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 ∈ 𝐽)
111	110	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑅 ∈ 𝑋 ∧ 𝑣 ∈ 𝐾)) → 𝑋 ∈ 𝐽)
112		simprr 796	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑅 ∈ 𝑋 ∧ 𝑣 ∈ 𝐾)) → 𝑣 ∈ 𝐾)
113		txopn 21405	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑋 ∈ 𝐽 ∧ 𝑣 ∈ 𝐾)) → (𝑋 × 𝑣) ∈ (𝐽 ×t 𝐾))
114	107, 108, 111, 112, 113	syl22anc 1327	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑅 ∈ 𝑋 ∧ 𝑣 ∈ 𝐾)) → (𝑋 × 𝑣) ∈ (𝐽 ×t 𝐾))
115		eleq2 2690	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝑋 × 𝑣) → (⟨𝑅, 𝑆⟩ ∈ 𝑤 ↔ ⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑣)))
116		eleq2 2690	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (𝑋 × 𝑣) → ((𝐻‘𝑘) ∈ 𝑤 ↔ (𝐻‘𝑘) ∈ (𝑋 × 𝑣)))
117	116	rexralbidv 3058	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝑋 × 𝑣) → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤 ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ (𝑋 × 𝑣)))
118	115, 117	imbi12d 334	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝑋 × 𝑣) → ((⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤) ↔ (⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑣) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ (𝑋 × 𝑣))))
119	118	rspcv 3305	. . . . . . . . . . 11 ⊢ ((𝑋 × 𝑣) ∈ (𝐽 ×t 𝐾) → (∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤) → (⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑣) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ (𝑋 × 𝑣))))
120	114, 119	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑅 ∈ 𝑋 ∧ 𝑣 ∈ 𝐾)) → (∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤) → (⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑣) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ (𝑋 × 𝑣))))
121		opelxpi 5148	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝑣) → ⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑣))
122	121	ex 450	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ 𝑋 → (𝑆 ∈ 𝑣 → ⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑣)))
123	122	ad2antrl 764	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑅 ∈ 𝑋 ∧ 𝑣 ∈ 𝐾)) → (𝑆 ∈ 𝑣 → ⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑣)))
124	50	eleq1d 2686	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ 𝑍 → ((𝐻‘𝑘) ∈ (𝑋 × 𝑣) ↔ ⟨(𝐹‘𝑘), (𝐺‘𝑘)⟩ ∈ (𝑋 × 𝑣)))
125		opelxp2 5151	. . . . . . . . . . . . . . . 16 ⊢ (⟨(𝐹‘𝑘), (𝐺‘𝑘)⟩ ∈ (𝑋 × 𝑣) → (𝐺‘𝑘) ∈ 𝑣)
126	124, 125	syl6bi 243	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ 𝑍 → ((𝐻‘𝑘) ∈ (𝑋 × 𝑣) → (𝐺‘𝑘) ∈ 𝑣))
127	43, 126	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → ((𝐻‘𝑘) ∈ (𝑋 × 𝑣) → (𝐺‘𝑘) ∈ 𝑣))
128	127	ralimdva 2962	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ 𝑍 → (∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ (𝑋 × 𝑣) → ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣))
129	128	reximia 3009	. . . . . . . . . . . 12 ⊢ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ (𝑋 × 𝑣) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)
130	129	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑅 ∈ 𝑋 ∧ 𝑣 ∈ 𝐾)) → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ (𝑋 × 𝑣) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣))
131	123, 130	imim12d 81	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑅 ∈ 𝑋 ∧ 𝑣 ∈ 𝐾)) → ((⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑣) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ (𝑋 × 𝑣)) → (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)))
132	120, 131	syld 47	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑅 ∈ 𝑋 ∧ 𝑣 ∈ 𝐾)) → (∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤) → (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)))
133	132	anassrs 680	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑅 ∈ 𝑋) ∧ 𝑣 ∈ 𝐾) → (∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤) → (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)))
134	133	ralrimdva 2969	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑅 ∈ 𝑋) → (∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤) → ∀𝑣 ∈ 𝐾 (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)))
135	134	adantrr 753	. . . . . 6 ⊢ ((𝜑 ∧ (𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝑌)) → (∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤) → ∀𝑣 ∈ 𝐾 (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)))
136	106, 135	jcad 555	. . . . 5 ⊢ ((𝜑 ∧ (𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝑌)) → (∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤) → (∀𝑢 ∈ 𝐽 (𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ ∀𝑣 ∈ 𝐾 (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣))))
137	76, 136	impbid 202	. . . 4 ⊢ ((𝜑 ∧ (𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝑌)) → ((∀𝑢 ∈ 𝐽 (𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ ∀𝑣 ∈ 𝐾 (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)) ↔ ∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤)))
138	137	pm5.32da 673	. . 3 ⊢ (𝜑 → (((𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝑌) ∧ (∀𝑢 ∈ 𝐽 (𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ ∀𝑣 ∈ 𝐾 (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣))) ↔ ((𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝑌) ∧ ∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤))))
139		opelxp 5146	. . . 4 ⊢ (⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑌) ↔ (𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝑌))
140	139	anbi1i 731	. . 3 ⊢ ((⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑌) ∧ ∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤)) ↔ ((𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝑌) ∧ ∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤)))
141	138, 140	syl6bbr 278	. 2 ⊢ (𝜑 → (((𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝑌) ∧ (∀𝑢 ∈ 𝐽 (𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ ∀𝑣 ∈ 𝐾 (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣))) ↔ (⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑌) ∧ ∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤))))
142		txlm.m	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
143		txlm.f	. . . . 5 ⊢ (𝜑 → 𝐹:𝑍⟶𝑋)
144		eqidd 2623	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐹‘𝑘))
145	6, 41, 142, 143, 144	lmbrf 21064	. . . 4 ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑅 ↔ (𝑅 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢))))
146		txlm.g	. . . . 5 ⊢ (𝜑 → 𝐺:𝑍⟶𝑌)
147		eqidd 2623	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (𝐺‘𝑘))
148	9, 41, 142, 146, 147	lmbrf 21064	. . . 4 ⊢ (𝜑 → (𝐺(⇝𝑡‘𝐾)𝑆 ↔ (𝑆 ∈ 𝑌 ∧ ∀𝑣 ∈ 𝐾 (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣))))
149	145, 148	anbi12d 747	. . 3 ⊢ (𝜑 → ((𝐹(⇝𝑡‘𝐽)𝑅 ∧ 𝐺(⇝𝑡‘𝐾)𝑆) ↔ ((𝑅 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢)) ∧ (𝑆 ∈ 𝑌 ∧ ∀𝑣 ∈ 𝐾 (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)))))
150		an4 865	. . 3 ⊢ (((𝑅 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢)) ∧ (𝑆 ∈ 𝑌 ∧ ∀𝑣 ∈ 𝐾 (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣))) ↔ ((𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝑌) ∧ (∀𝑢 ∈ 𝐽 (𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ ∀𝑣 ∈ 𝐾 (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣))))
151	149, 150	syl6bb 276	. 2 ⊢ (𝜑 → ((𝐹(⇝𝑡‘𝐽)𝑅 ∧ 𝐺(⇝𝑡‘𝐾)𝑆) ↔ ((𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝑌) ∧ (∀𝑢 ∈ 𝐽 (𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ ∀𝑣 ∈ 𝐾 (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)))))
152		txtopon 21394	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
153	6, 9, 152	syl2anc 693	. . 3 ⊢ (𝜑 → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
154	143	ffvelrnda 6359	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ 𝑋)
155	146	ffvelrnda 6359	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐺‘𝑛) ∈ 𝑌)
156		opelxpi 5148	. . . . 5 ⊢ (((𝐹‘𝑛) ∈ 𝑋 ∧ (𝐺‘𝑛) ∈ 𝑌) → ⟨(𝐹‘𝑛), (𝐺‘𝑛)⟩ ∈ (𝑋 × 𝑌))
157	154, 155, 156	syl2anc 693	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ⟨(𝐹‘𝑛), (𝐺‘𝑛)⟩ ∈ (𝑋 × 𝑌))
158	157, 48	fmptd 6385	. . 3 ⊢ (𝜑 → 𝐻:𝑍⟶(𝑋 × 𝑌))
159		eqidd 2623	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = (𝐻‘𝑘))
160	153, 41, 142, 158, 159	lmbrf 21064	. 2 ⊢ (𝜑 → (𝐻(⇝𝑡‘(𝐽 ×t 𝐾))⟨𝑅, 𝑆⟩ ↔ (⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑌) ∧ ∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤))))
161	141, 151, 160	3bitr4d 300	1 ⊢ (𝜑 → ((𝐹(⇝𝑡‘𝐽)𝑅 ∧ 𝐺(⇝𝑡‘𝐾)𝑆) ↔ 𝐻(⇝𝑡‘(𝐽 ×t 𝐾))⟨𝑅, 𝑆⟩))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ⊆ wss 3574  ⟨cop 4183   class class class wbr 4653   ↦ cmpt 4729   × cxp 5112  ran crn 5115  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  ℤcz 11377  ℤ_≥cuz 11687  topGenctg 16098  Topctop 20698  TopOnctopon 20715  ⇝_𝑡clm 21030   ×_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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-pre-lttri 10010  ax-pre-lttrn 10011

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-nel 2898  df-ral 2917  df-rex 2918  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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  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-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-1st 7168  df-2nd 7169  df-er 7742  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-neg 10269  df-z 11378  df-uz 11688  df-topgen 16104  df-top 20699  df-topon 20716  df-bases 20750  df-lm 21033  df-tx 21365

This theorem is referenced by:  lmcn2  21452