Theorem txlm 12290

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 2533	. . . . . . . 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 2534	. . . . . . . . 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 2469	. . . . . . . 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 14	. . . . . . 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 503	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑤 ∈ (𝐽 ×t 𝐾) ∧ ⟨𝑅, 𝑆⟩ ∈ 𝑤)) → 𝑤 ∈ (𝐽 ×t 𝐾))
6		txlm.j	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
7		topontop 12024	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
8	6, 7	syl 14	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐽 ∈ Top)
9		txlm.k	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
10		topontop 12024	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
11	9, 10	syl 14	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐾 ∈ Top)
12		eqid 2115	. . . . . . . . . . . . . . . . . 18 ⊢ ran (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣)) = ran (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣))
13	12	txval 12266	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ×t 𝐾) = (topGen‘ran (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣))))
14	8, 11, 13	syl2anc 406	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐽 ×t 𝐾) = (topGen‘ran (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣))))
15	14	adantr 272	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑤 ∈ (𝐽 ×t 𝐾) ∧ ⟨𝑅, 𝑆⟩ ∈ 𝑤)) → (𝐽 ×t 𝐾) = (topGen‘ran (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣))))
16	5, 15	eleqtrd 2193	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑤 ∈ (𝐽 ×t 𝐾) ∧ ⟨𝑅, 𝑆⟩ ∈ 𝑤)) → 𝑤 ∈ (topGen‘ran (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣))))
17		simprr 504	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑤 ∈ (𝐽 ×t 𝐾) ∧ ⟨𝑅, 𝑆⟩ ∈ 𝑤)) → ⟨𝑅, 𝑆⟩ ∈ 𝑤)
18		tg2 12072	. . . . . . . . . . . . . 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 406	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑤 ∈ (𝐽 ×t 𝐾) ∧ ⟨𝑅, 𝑆⟩ ∈ 𝑤)) → ∃𝑡 ∈ ran (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣))(⟨𝑅, 𝑆⟩ ∈ 𝑡 ∧ 𝑡 ⊆ 𝑤))
20		vex 2660	. . . . . . . . . . . . . . . 16 ⊢ 𝑢 ∈ V
21		vex 2660	. . . . . . . . . . . . . . . 16 ⊢ 𝑣 ∈ V
22	20, 21	xpex 4614	. . . . . . . . . . . . . . 15 ⊢ (𝑢 × 𝑣) ∈ V
23	22	rgen2w 2462	. . . . . . . . . . . . . 14 ⊢ ∀𝑢 ∈ 𝐽 ∀𝑣 ∈ 𝐾 (𝑢 × 𝑣) ∈ V
24		eqid 2115	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣)) = (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣))
25		eleq2 2178	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = (𝑢 × 𝑣) → (⟨𝑅, 𝑆⟩ ∈ 𝑡 ↔ ⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣)))
26		sseq1 3086	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = (𝑢 × 𝑣) → (𝑡 ⊆ 𝑤 ↔ (𝑢 × 𝑣) ⊆ 𝑤))
27	25, 26	anbi12d 462	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = (𝑢 × 𝑣) → ((⟨𝑅, 𝑆⟩ ∈ 𝑡 ∧ 𝑡 ⊆ 𝑤) ↔ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)))
28	24, 27	rexrnmpo 5840	. . . . . . . . . . . . . 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 7	. . . . . . . . . . . . 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 121	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑤 ∈ (𝐽 ×t 𝐾) ∧ ⟨𝑅, 𝑆⟩ ∈ 𝑤)) → ∃𝑢 ∈ 𝐽 ∃𝑣 ∈ 𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤))
31	30	ex 114	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑤 ∈ (𝐽 ×t 𝐾) ∧ ⟨𝑅, 𝑆⟩ ∈ 𝑤) → ∃𝑢 ∈ 𝐽 ∃𝑣 ∈ 𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)))
32		r19.29 2543	. . . . . . . . . . . . 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 2543	. . . . . . . . . . . . . . 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 503	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑣 ∈ 𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) → ⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣))
35		opelxp 4529	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ↔ (𝑅 ∈ 𝑢 ∧ 𝑆 ∈ 𝑣))
36	34, 35	sylib 121	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑣 ∈ 𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) → (𝑅 ∈ 𝑢 ∧ 𝑆 ∈ 𝑣))
37		pm2.27 40	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑅 ∈ 𝑢 → ((𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢))
38		pm2.27 40	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑆 ∈ 𝑣 → ((𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣))
39	37, 38	im2anan9 570	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((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 14	. . . . . . . . . . . . . . . . . . 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)(𝐹‘𝑘) ∈ 𝑢 ∧ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)))
41		txlm.z	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑍 = (ℤ≥‘𝑀)
42	41	rexanuz2 10655	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∃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 9245	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → 𝑘 ∈ 𝑍)
44		opelxpi 4531	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝐹‘𝑘) ∈ 𝑢 ∧ (𝐺‘𝑘) ∈ 𝑣) → ⟨(𝐹‘𝑘), (𝐺‘𝑘)⟩ ∈ (𝑢 × 𝑣))
45		txlm.h	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ ⟨(𝐹‘𝑛), (𝐺‘𝑛)⟩)
46		fveq2 5375	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑛 = 𝑘 → (𝐹‘𝑛) = (𝐹‘𝑘))
47		fveq2 5375	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑛 = 𝑘 → (𝐺‘𝑛) = (𝐺‘𝑘))
48	46, 47	opeq12d 3679	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑛 = 𝑘 → ⟨(𝐹‘𝑛), (𝐺‘𝑛)⟩ = ⟨(𝐹‘𝑘), (𝐺‘𝑘)⟩)
49		simpr 109	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑣 ∈ 𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ 𝑍)
50		txlm.f	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝜑 → 𝐹:𝑍⟶𝑋)
51	50	ad4antr 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑣 ∈ 𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) ∧ 𝑘 ∈ 𝑍) → 𝐹:𝑍⟶𝑋)
52	51, 49	ffvelrnd 5510	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑣 ∈ 𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ 𝑋)
53		txlm.g	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝜑 → 𝐺:𝑍⟶𝑌)
54	53	ad4antr 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑣 ∈ 𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) ∧ 𝑘 ∈ 𝑍) → 𝐺:𝑍⟶𝑌)
55	54, 49	ffvelrnd 5510	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑣 ∈ 𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ 𝑌)
56		opexg 4110	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝐹‘𝑘) ∈ 𝑋 ∧ (𝐺‘𝑘) ∈ 𝑌) → ⟨(𝐹‘𝑘), (𝐺‘𝑘)⟩ ∈ V)
57	52, 55, 56	syl2anc 406	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑣 ∈ 𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) ∧ 𝑘 ∈ 𝑍) → ⟨(𝐹‘𝑘), (𝐺‘𝑘)⟩ ∈ V)
58	45, 48, 49, 57	fvmptd3 5468	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑣 ∈ 𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = ⟨(𝐹‘𝑘), (𝐺‘𝑘)⟩)
59	58	eleq1d 2183	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑣 ∈ 𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) ∧ 𝑘 ∈ 𝑍) → ((𝐻‘𝑘) ∈ (𝑢 × 𝑣) ↔ ⟨(𝐹‘𝑘), (𝐺‘𝑘)⟩ ∈ (𝑢 × 𝑣)))
60	44, 59	syl5ibr 155	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑣 ∈ 𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) ∧ 𝑘 ∈ 𝑍) → (((𝐹‘𝑘) ∈ 𝑢 ∧ (𝐺‘𝑘) ∈ 𝑣) → (𝐻‘𝑘) ∈ (𝑢 × 𝑣)))
61		simplrr 508	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑣 ∈ 𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) ∧ 𝑘 ∈ 𝑍) → (𝑢 × 𝑣) ⊆ 𝑤)
62	61	sseld 3062	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑣 ∈ 𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) ∧ 𝑘 ∈ 𝑍) → ((𝐻‘𝑘) ∈ (𝑢 × 𝑣) → (𝐻‘𝑘) ∈ 𝑤))
63	60, 62	syld 45	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑣 ∈ 𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) ∧ 𝑘 ∈ 𝑍) → (((𝐹‘𝑘) ∈ 𝑢 ∧ (𝐺‘𝑘) ∈ 𝑣) → (𝐻‘𝑘) ∈ 𝑤))
64	43, 63	sylan2 282	. . . . . . . . . . . . . . . . . . . . . . 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))
65	64	anassrs 395	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((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))
66	65	ralimdva 2473	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((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))
67	66	reximdva 2508	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((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))
68	42, 67	syl5bir 152	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((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))
69	40, 68	syld 45	. . . . . . . . . . . . . . . . . 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))
70	69	ex 114	. . . . . . . . . . . . . . . . 17 ⊢ (((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)))
71	70	impcomd 253	. . . . . . . . . . . . . . . 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))
72	71	rexlimdva 2523	. . . . . . . . . . . . . . 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))
73	33, 72	syl5 32	. . . . . . . . . . . . . 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))
74	73	rexlimdva 2523	. . . . . . . . . . . . 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))
75	32, 74	syl5 32	. . . . . . . . . . . 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))
76	75	expcomd 1400	. . . . . . . . . . 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)))
77	31, 76	syld 45	. . . . . . . . . 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)))
78	77	expdimp 257	. . . . . . . . 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)))
79	78	com23 78	. . . . . . . 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)))
80	79	ralrimdva 2486	. . . . . . 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)))
81	4, 80	syl5 32	. . . . . 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)))
82	81	adantr 272	. . . . 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)))
83	8	adantr 272	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑆 ∈ 𝑌 ∧ 𝑢 ∈ 𝐽)) → 𝐽 ∈ Top)
84	11	adantr 272	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑆 ∈ 𝑌 ∧ 𝑢 ∈ 𝐽)) → 𝐾 ∈ Top)
85		simprr 504	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑆 ∈ 𝑌 ∧ 𝑢 ∈ 𝐽)) → 𝑢 ∈ 𝐽)
86		toponmax 12035	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 ∈ 𝐾)
87	9, 86	syl 14	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 ∈ 𝐾)
88	87	adantr 272	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑆 ∈ 𝑌 ∧ 𝑢 ∈ 𝐽)) → 𝑌 ∈ 𝐾)
89		txopn 12276	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑢 ∈ 𝐽 ∧ 𝑌 ∈ 𝐾)) → (𝑢 × 𝑌) ∈ (𝐽 ×t 𝐾))
90	83, 84, 85, 88, 89	syl22anc 1200	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑆 ∈ 𝑌 ∧ 𝑢 ∈ 𝐽)) → (𝑢 × 𝑌) ∈ (𝐽 ×t 𝐾))
91		eleq2 2178	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝑢 × 𝑌) → (⟨𝑅, 𝑆⟩ ∈ 𝑤 ↔ ⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑌)))
92		eleq2 2178	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (𝑢 × 𝑌) → ((𝐻‘𝑘) ∈ 𝑤 ↔ (𝐻‘𝑘) ∈ (𝑢 × 𝑌)))
93	92	rexralbidv 2435	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝑢 × 𝑌) → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤 ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ (𝑢 × 𝑌)))
94	91, 93	imbi12d 233	. . . . . . . . . . . 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))))
95	94	rspcv 2756	. . . . . . . . . . 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))))
96	90, 95	syl 14	. . . . . . . . . 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))))
97		simprl 503	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑆 ∈ 𝑌 ∧ 𝑢 ∈ 𝐽)) → 𝑆 ∈ 𝑌)
98		opelxpi 4531	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ 𝑢 ∧ 𝑆 ∈ 𝑌) → ⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑌))
99	97, 98	sylan2 282	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ 𝑢 ∧ (𝜑 ∧ (𝑆 ∈ 𝑌 ∧ 𝑢 ∈ 𝐽))) → ⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑌))
100	99	expcom 115	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑆 ∈ 𝑌 ∧ 𝑢 ∈ 𝐽)) → (𝑅 ∈ 𝑢 → ⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑌)))
101		simpr 109	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ 𝑍)
102	50	ffvelrnda 5509	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ 𝑋)
103	53	ffvelrnda 5509	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ 𝑌)
104	102, 103, 56	syl2anc 406	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ⟨(𝐹‘𝑘), (𝐺‘𝑘)⟩ ∈ V)
105	45, 48, 101, 104	fvmptd3 5468	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = ⟨(𝐹‘𝑘), (𝐺‘𝑘)⟩)
106	105	eleq1d 2183	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐻‘𝑘) ∈ (𝑢 × 𝑌) ↔ ⟨(𝐹‘𝑘), (𝐺‘𝑘)⟩ ∈ (𝑢 × 𝑌)))
107		opelxp1 4533	. . . . . . . . . . . . . . . . 17 ⊢ (⟨(𝐹‘𝑘), (𝐺‘𝑘)⟩ ∈ (𝑢 × 𝑌) → (𝐹‘𝑘) ∈ 𝑢)
108	106, 107	syl6bi 162	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐻‘𝑘) ∈ (𝑢 × 𝑌) → (𝐹‘𝑘) ∈ 𝑢))
109	43, 108	sylan2 282	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘𝑗))) → ((𝐻‘𝑘) ∈ (𝑢 × 𝑌) → (𝐹‘𝑘) ∈ 𝑢))
110	109	anassrs 395	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → ((𝐻‘𝑘) ∈ (𝑢 × 𝑌) → (𝐹‘𝑘) ∈ 𝑢))
111	110	ralimdva 2473	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ (𝑢 × 𝑌) → ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢))
112	111	reximdva 2508	. . . . . . . . . . . 12 ⊢ (𝜑 → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ (𝑢 × 𝑌) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢))
113	112	adantr 272	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑆 ∈ 𝑌 ∧ 𝑢 ∈ 𝐽)) → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ (𝑢 × 𝑌) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢))
114	100, 113	imim12d 74	. . . . . . . . . 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)))
115	96, 114	syld 45	. . . . . . . . 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)))
116	115	anassrs 395	. . . . . . . 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)))
117	116	ralrimdva 2486	. . . . . . 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)))
118	117	adantrl 467	. . . . . 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)))
119	8	adantr 272	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑅 ∈ 𝑋 ∧ 𝑣 ∈ 𝐾)) → 𝐽 ∈ Top)
120	11	adantr 272	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑅 ∈ 𝑋 ∧ 𝑣 ∈ 𝐾)) → 𝐾 ∈ Top)
121		toponmax 12035	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
122	6, 121	syl 14	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 ∈ 𝐽)
123	122	adantr 272	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑅 ∈ 𝑋 ∧ 𝑣 ∈ 𝐾)) → 𝑋 ∈ 𝐽)
124		simprr 504	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑅 ∈ 𝑋 ∧ 𝑣 ∈ 𝐾)) → 𝑣 ∈ 𝐾)
125		txopn 12276	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑋 ∈ 𝐽 ∧ 𝑣 ∈ 𝐾)) → (𝑋 × 𝑣) ∈ (𝐽 ×t 𝐾))
126	119, 120, 123, 124, 125	syl22anc 1200	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑅 ∈ 𝑋 ∧ 𝑣 ∈ 𝐾)) → (𝑋 × 𝑣) ∈ (𝐽 ×t 𝐾))
127		eleq2 2178	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝑋 × 𝑣) → (⟨𝑅, 𝑆⟩ ∈ 𝑤 ↔ ⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑣)))
128		eleq2 2178	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (𝑋 × 𝑣) → ((𝐻‘𝑘) ∈ 𝑤 ↔ (𝐻‘𝑘) ∈ (𝑋 × 𝑣)))
129	128	rexralbidv 2435	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝑋 × 𝑣) → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤 ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ (𝑋 × 𝑣)))
130	127, 129	imbi12d 233	. . . . . . . . . . . 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))))
131	130	rspcv 2756	. . . . . . . . . . 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))))
132	126, 131	syl 14	. . . . . . . . . 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))))
133		opelxpi 4531	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝑣) → ⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑣))
134	133	ex 114	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ 𝑋 → (𝑆 ∈ 𝑣 → ⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑣)))
135	134	ad2antrl 479	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑅 ∈ 𝑋 ∧ 𝑣 ∈ 𝐾)) → (𝑆 ∈ 𝑣 → ⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑣)))
136	105	eleq1d 2183	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐻‘𝑘) ∈ (𝑋 × 𝑣) ↔ ⟨(𝐹‘𝑘), (𝐺‘𝑘)⟩ ∈ (𝑋 × 𝑣)))
137		opelxp2 4534	. . . . . . . . . . . . . . . . 17 ⊢ (⟨(𝐹‘𝑘), (𝐺‘𝑘)⟩ ∈ (𝑋 × 𝑣) → (𝐺‘𝑘) ∈ 𝑣)
138	136, 137	syl6bi 162	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐻‘𝑘) ∈ (𝑋 × 𝑣) → (𝐺‘𝑘) ∈ 𝑣))
139	43, 138	sylan2 282	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘𝑗))) → ((𝐻‘𝑘) ∈ (𝑋 × 𝑣) → (𝐺‘𝑘) ∈ 𝑣))
140	139	anassrs 395	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → ((𝐻‘𝑘) ∈ (𝑋 × 𝑣) → (𝐺‘𝑘) ∈ 𝑣))
141	140	ralimdva 2473	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ (𝑋 × 𝑣) → ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣))
142	141	reximdva 2508	. . . . . . . . . . . 12 ⊢ (𝜑 → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ (𝑋 × 𝑣) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣))
143	142	adantr 272	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑅 ∈ 𝑋 ∧ 𝑣 ∈ 𝐾)) → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ (𝑋 × 𝑣) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣))
144	135, 143	imim12d 74	. . . . . . . . . 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)))
145	132, 144	syld 45	. . . . . . . . 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)))
146	145	anassrs 395	. . . . . . . 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)))
147	146	ralrimdva 2486	. . . . . . 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)))
148	147	adantrr 468	. . . . . 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)))
149	118, 148	jcad 303	. . . . 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))))
150	82, 149	impbid 128	. . . 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)))
151	150	pm5.32da 445	. . 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))))
152		opelxp 4529	. . . 4 ⊢ (⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑌) ↔ (𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝑌))
153	152	anbi1i 451	. . 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)))
154	151, 153	syl6bbr 197	. 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))))
155		txlm.m	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
156		eqidd 2116	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐹‘𝑘))
157	6, 41, 155, 50, 156	lmbrf 12226	. . . 4 ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑅 ↔ (𝑅 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢))))
158		eqidd 2116	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (𝐺‘𝑘))
159	9, 41, 155, 53, 158	lmbrf 12226	. . . 4 ⊢ (𝜑 → (𝐺(⇝𝑡‘𝐾)𝑆 ↔ (𝑆 ∈ 𝑌 ∧ ∀𝑣 ∈ 𝐾 (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣))))
160	157, 159	anbi12d 462	. . 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)))))
161		an4 558	. . 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))))
162	160, 161	syl6bb 195	. 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)))))
163		txtopon 12273	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
164	6, 9, 163	syl2anc 406	. . 3 ⊢ (𝜑 → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
165	50	ffvelrnda 5509	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ 𝑋)
166	53	ffvelrnda 5509	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐺‘𝑛) ∈ 𝑌)
167	165, 166	opelxpd 4532	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ⟨(𝐹‘𝑛), (𝐺‘𝑛)⟩ ∈ (𝑋 × 𝑌))
168	167, 45	fmptd 5528	. . 3 ⊢ (𝜑 → 𝐻:𝑍⟶(𝑋 × 𝑌))
169		eqidd 2116	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = (𝐻‘𝑘))
170	164, 41, 155, 168, 169	lmbrf 12226	. 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))))
171	154, 162, 170	3bitr4d 219	1 ⊢ (𝜑 → ((𝐹(⇝𝑡‘𝐽)𝑅 ∧ 𝐺(⇝𝑡‘𝐾)𝑆) ↔ 𝐻(⇝𝑡‘(𝐽 ×t 𝐾))⟨𝑅, 𝑆⟩))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1314   ∈ wcel 1463  ∀wral 2390  ∃wrex 2391  Vcvv 2657   ⊆ wss 3037  ⟨cop 3496   class class class wbr 3895   ↦ cmpt 3949   × cxp 4497  ran crn 4500  ⟶wf 5077  ‘cfv 5081  (class class class)co 5728   ∈ cmpo 5730  ℤcz 8958  ℤ_≥cuz 9228  topGenctg 11978  Topctop 12007  TopOnctopon 12020  ⇝_𝑡clm 12199   ×_t ctx 12263

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4003  ax-sep 4006  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-cnex 7636  ax-resscn 7637  ax-1cn 7638  ax-1re 7639  ax-icn 7640  ax-addcl 7641  ax-addrcl 7642  ax-mulcl 7643  ax-addcom 7645  ax-addass 7647  ax-distr 7649  ax-i2m1 7650  ax-0lt1 7651  ax-0id 7653  ax-rnegex 7654  ax-cnre 7656  ax-pre-ltirr 7657  ax-pre-ltwlin 7658  ax-pre-lttrn 7659  ax-pre-apti 7660  ax-pre-ltadd 7661

This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-nel 2378  df-ral 2395  df-rex 2396  df-reu 2397  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-if 3441  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-id 4175  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089  df-riota 5684  df-ov 5731  df-oprab 5732  df-mpo 5733  df-1st 5992  df-2nd 5993  df-pm 6499  df-pnf 7726  df-mnf 7727  df-xr 7728  df-ltxr 7729  df-le 7730  df-sub 7858  df-neg 7859  df-inn 8631  df-n0 8882  df-z 8959  df-uz 9229  df-topgen 11984  df-top 12008  df-topon 12021  df-bases 12053  df-lm 12202  df-tx 12264

This theorem is referenced by:  lmcn2  12291